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" }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "f(x)=2^x" "6#/-%\"fG6#%\"xG)\"\"#F'" } {TEXT -1 39 ", the graph of f indicates that f is a " }{TEXT 262 19 "o ne-to-one function" }{TEXT -1 73 ", since every horizontal line meets \+ the graph at no more than one point. " }}{PARA 0 "" 0 "" {TEXT -1 37 " Thus, given any positive real number " }{TEXT 384 1 "y" }{TEXT -1 28 " , we can find a real number " }{TEXT 383 1 "x" }{TEXT -1 12 " such tha t " }{XPPEDIT 18 0 "2^x=y" "6#/)\"\"#%\"xG%\"yG" }{TEXT -1 16 ". 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" }}{PARA 256 "" 0 "" {TEXT 387 13 "_____________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Here are some examples: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[2]*8=3" "6#/*&&%$logG6#\"\"#\"\"\"\"\")F)\"\"$ " }{TEXT -1 18 " because " }{XPPEDIT 18 0 "2^3=8" "6#/*$\"\"# \"\"$\"\")" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "log[2]*32=5" "6#/*&&%$logG6#\"\"#\"\"\"\"#KF)\"\"&" } {TEXT -1 20 " because " }{XPPEDIT 18 0 "2^5=32" "6#/*$\"\"# \"\"&\"#K" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "log[2]*2=1" "6#/*&&%$logG6#\"\"#\"\"\"F(F)F)" }{TEXT -1 21 " because " }{XPPEDIT 18 0 "2^``(1)=2" "6#/)\"\"#-%! G6#\"\"\"F%" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "log[2]*1=0" "6#/*&&%$logG6#\"\"#\"\"\"F)F)\"\"!" } {TEXT -1 21 " because " }{XPPEDIT 18 0 "2^0=1" "6#/*$\"\"# \"\"!\"\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "log[2]*``(1/2) = -1;" "6#/*&&%$logG6#\"\"#\"\"\"-%!G6#* &F)F)F(!\"\"F),$F)F." }{TEXT -1 21 " because " }{XPPEDIT 18 0 "2^(-1)=1/2" "6#/)\"\"#,$\"\"\"!\"\"*&F'F'F%F(" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[2]*sqrt(2)=1/2 " "6#/*&&%$logG6#\"\"#\"\"\"-%%sqrtG6#F(F)*&F)F)F(!\"\"" }{TEXT -1 21 " because " }{XPPEDIT 18 0 "2^(1/2)=sqrt(2)" "6#/)\"\"#*& \"\"\"F'F%!\"\"-%%sqrtG6#F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 166 "For these examples we can find the logarithm to base 2 of the \+ given (input) number by expressing the number as a power of 2, and tak ing that power for the logarithm. " }}{PARA 0 "" 0 "" {TEXT -1 44 "At \+ present we do not have a way of finding " }{XPPEDIT 18 0 "log[2]*x" " 6#*&&%$logG6#\"\"#\"\"\"%\"xGF(" }{TEXT -1 21 " unless we recognize " }{TEXT 389 1 "x" }{TEXT -1 18 " as a power of 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The function " } {XPPEDIT 18 0 "log[2]" "6#&%$logG6#\"\"#" }{TEXT -1 30 " (logarithm to base 2) is the " }{TEXT 262 16 "inverse function" }{TEXT -1 43 " of f . Denoting this inverse function by f" }{XPPEDIT 18 0 "``^(-1)" "6#)% !G,$\"\"\"!\"\"" }{TEXT -1 12 ", we have f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 1 "(" }{TEXT 388 1 "x" }{TEXT -1 1 ") " }{XPPEDIT 18 0 "`` = log[2]*x" "6#/%!G*&&%$logG6#\"\"#\"\"\"%\"xGF* " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph of " } {XPPEDIT 18 0 "y=log[2]*x" "6#/%\"yG*&&%$logG6#\"\"#\"\"\"%\"xGF*" } {TEXT -1 89 " is the graph of this inverse function, and, as usual, it is the reflection in the line " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG " }{TEXT -1 18 " of the graph of " }{XPPEDIT 18 0 "y=2^x" "6#/%\"yG) \"\"#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 446 428 428 {PLOTDATA 2 "6eo-%'C URVESG6%7U7$$!3M+++++++W!#<$\"3-$*\\MqUhOZ!#>7$$!3/++](>#)*HUF*$\"38GN 5KQ.H`F-7$$!3e+]i;10#3%F*$\"30rs!f%=Y/fF-7$$!3f++D#3'o:RF*$\"3]U4eIl9E mF-7$$!3I++vv(=#[PF*$\"3?jZGJnsTuF-7$$!3Q+]77ua\"e$F*$\"3i*>1Z-$3`$)F- 7$$!3g+]iA<-FMF*$\"3%*fvtAgW(H*F-7$$!3I+]i@$>qE$F*$\"3@\"G>YT$zQ5!#=7$ $!3M+]i4Ya,JF*$\"3CtuYFK/l6FQ7$$!3b+]7x0gOHF*$\"38([OWFchI\"FQ7$$!3?++ ]l`$pw#F*$\"3]))yEH7;p9FQ7$$!39++vaT\\F*$\"3Ox`h\\F/`DFQ7$$!39++]\"p DRz\"F*$\"3u]k9ZD'Q)GFQ7$$!3B+++')H,X;F*$\"3gt3Gs^V(>$FQ7$$!3K+]i3@zr9 F*$\"3Q%*\\]o;M0OFQ7$$!3S+++\"Gi%=8F*$\"3UG,-G3i4SFQ7$$!39+]ie^B]6F*$ \"3)\\;pe&y^0XFQ7$$!3D++ve:V+**FQ$\"3\"G$4NXriM]FQ7$$!3E++]ig**G#)FQ$ \"3qhq5/i/`cFQ7$$!3g++D6l4%p'FQ$\"3%H:4)R#RwG'FQ7$$!3]-+]()**\\Q]FQ$\" 37I))o/IA_qFQ7$$!3i++D'\\+)=LFQ$\"3K?J6ZX+XzFQ7$$!3d0+vLqz@=FQ$\"33f<% zm.P\"))FQ7$$!3)y-+]#G#*\\?F-$\"3'**=h*))[\"*e)*FQ7$$\"39%*****H_Kl9FQ $\"38m+Ebe!p5\"F*7$$\"3M#***\\Z]S*4$FQ$\"3WM4twllR7F*7$$\"3)y**\\PHh/o %FQ$\"3]F*7$$\"3H**\\P%ynB 7\"F*$\"3OB&QD!)Rq<#F*7$$\"35++]TY@*G\"F*$\"3)frGEy\\RW#F*7$$\"3%*)*\\ PJX?Y9F*$\"3UhwC_B!\\s#F*7$$\"3P+](3?/.h\"F*$\"3#GlUv7iJ0$F*7$$\"3A*** *\\A;uqF*$\"3&pF0n28O$QF*7$$\"3A+++ uJZ+@F*$\"3YpnYe,])G%F*7$$\"3a***\\#oZ!fE#F*$\"3woM#z]f&4[F*7$$\"3#*** \\P0l'*HCF*$\"3-gwh&>4))Q&F*7$$\"3a+++.As!e#F*$\"3ajH)z)4R#)fF*7$$\"3S ++DX_]`FF*$\"3;$H`;(e`VnF*7$$\"3++++eW/3HF*$\"3`YT*G(3+1vF*7$$\"3m**\\ P\"\\:G2$F*$\"3**om&p1QTT)F*7$$\"3y*****H%*o;:$F*$\"3#e![RB.$o)))F*7$$ \"3!***\\i%RA0B$F*$\"3?9-h_J$F*$\"3YXx(y>&*Q&**F* 7$$\"3!**************R$F*$\"3O:$='Gjgb5!#;-%&COLORG6&%$RGBG$\"#$*!\"#$ \"\"!F\\\\lF[\\l-%*THICKNESSG6#\"\"#-F$6%7U7$F+F(7$F1F/7$F6F47$F;F97$F @F>7$FEFC7$FJFH7$FOFM7$FUFS7$FZFX7$FinFgn7$F^oF\\o7$FcoFao7$FhoFfo7$F] pF[p7$FbpF`p7$FgpFep7$F\\qFjp7$FaqF_q7$FfqFdq7$F[rFiq7$F`rF^r7$FerFcr7 $FjrFhr7$F_sF]s7$FdsFbs7$FisFgs7$F^tF\\t7$FctFat7$FhtFft7$F]uF[u7$FbuF `u7$FguFeu7$F\\vFju7$FavF_v7$FfvFdv7$F[wFiv7$F`wF^w7$FewFcw7$FjwFhw7$F _xF]x7$FdxFbx7$FixFgx7$F^yF\\y7$FcyFay7$FhyFfy7$F]zF[z7$FbzF`z7$FgzFez 7$F\\[lFjz7$Fa[lF_[l-Fe[l6&Fg[l$\"\"$!\"\"$\"\"\"F[`l$\"#')Fj[lF]\\l-F $6&7$7$F[\\l$F]`lF\\\\l7$Fd`l$F`\\lF\\\\l-%'COLOURG6&Fg[lF\\\\lF\\\\lF \\\\l-%'SYMBOLG6$%'CIRCLEG\"#:-%&STYLEG6#%&POINTG-F$6&Fb`l-Fe[l6&Fg[lF [\\lFd`lF[\\l-F[al6$F]al\"#5F_al-F$6&Fb`lFeal-F[al6$%(DIAMONDGFialF_al -F$6&Fb`lFeal-F[al6$%&CROSSGFialF_al-F$6&7S7$F(F(7$$!3@nmmEk*47%F*Fibl 7$$!3wLLL4pByQF*F\\cl7$$!3?nmmU%G_g$F*F_cl7$$!30nmm)H5/L$F*Fbcl7$$!3sL LL<$)*o0$F*Fecl7$$!3-nmm)pF*Fadl7$$!3MLLLh!*4?\"F*Fjdl7$$!3<1+++Or:#*FQF]el7$$!3)Q+++;gUa'FQF`el7$$ !3rqmm1XK=TFQFcel7$$!3WSLLtBlL7FQFfel7$$\"3/mmm1L/57FQFiel7$$\"3-&**** *>Bk_SFQF\\fl7$$\"3Gbmm1T#)olFQF_fl7$$\"3W(*****>VZH$*FQFbfl7$$\"3s*** **fhEe>\"F*Fefl7$$\"3mmmmmM6q9F*Fhfl7$$\"3Dmmm=W*>s\"F*F[gl7$$\"3/LLL` ?o$*>F*F^gl7$$\"3kKLL\\\"*)eF#F*Fagl7$$\"3_*****fD^:_#F*Fdgl7$$\"3-LLL hI(oy#F*Fggl7$$\"3-*****zWw41$F*Fjgl7$$\"3Q*****f8L\"HLF*F]hl7$$\"3;** ****>&)e)e$F*F`hl7$$\"3A*****z[pm(QF*Fchl7$$\"3qkmm]I_NTF*Ffhl7$$\"3Y+ ++![/>T%F*Fihl7$$\"3SKLLLrMiYF*F\\il7$$\"3y*****R5Zh$\\F*F_il7$$\"3'\\ mm'Q:x$>&F*Fbil7$$\"3I+++/L1jaF*Feil7$$\"37mmmE_MEdF*Fhil7$$\"3M)***** **p(>+'F*F[jl7$$\"3-MLLdJWniF*F^jl7$$\"3hKLLX1#*QlF*Fajl7$$\"3 " 0 "" {MPLTEXT 1 0 1 ":" }}}{PARA 0 "" 0 "" {TEXT -1 46 "More, generally, given a positiv e real number " }{TEXT 390 1 "a" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "a<>1" "6#0%\"aG\"\"\"" }{TEXT -1 24 ", the function f where " } {XPPEDIT 18 0 "f(x)=a^x" "6#/-%\"fG6#%\"xG)%\"aGF'" }{TEXT -1 54 " is \+ a one-to-one function and has the inverse function" }}{PARA 0 "" 0 "" {TEXT -1 3 " f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" } {TEXT -1 11 " given by f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\" \"" }{TEXT -1 1 "(" }{TEXT 391 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "` `=log[a]*x" "6#/%!G*&&%$logG6#%\"aG\"\"\"%\"xGF*" }{TEXT -1 7 " where \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=log[a]*x" "6#/% \"yG*&&%$logG6#%\"aG\"\"\"%\"xGF*" }{TEXT -1 18 " exactly when " } {XPPEDIT 18 0 "a^x=y" "6#/)%\"aG%\"xG%\"yG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT 392 13 "_____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 393 1 " a" }{TEXT -1 29 " > 1, the function f, where " }{XPPEDIT 18 0 "f(x)=a ^x" "6#/-%\"fG6#%\"xG)%\"aGF'" }{TEXT -1 8 ", is an " }{TEXT 262 19 "i ncreasing function" }{TEXT -1 37 ", that is, whenever the input number " }{TEXT 394 1 "x" }{TEXT -1 18 " increases from " }{XPPEDIT 18 0 " x = x[1];" "6#/%\"xG&F$6#\"\"\"" }{TEXT -1 5 " to " }{XPPEDIT 18 0 "x = x[2];" "6#/%\"xG&F$6#\"\"#" }{TEXT -1 45 ", the output also increa ses from the value " }{XPPEDIT 18 0 "y[1] = a^x[1];" "6#/&%\"yG6#\"\" \")%\"aG&%\"xG6#F'" }{TEXT -1 15 " to the value " }{XPPEDIT 18 0 "y[2 ] = a^x[2];" "6#/&%\"yG6#\"\"#)%\"aG&%\"xG6#F'" }{TEXT -1 47 ", corres ponding to the fact that the graph of " }{XPPEDIT 18 0 "y=a^x" "6#/% \"yG)%\"aG%\"xG" }{TEXT -1 42 " always goes upwards from left to righ t. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 1 " \+ " }{GLPLOT2D 326 314 314 {PLOTDATA 2 "62-%'CURVESG6%7S7$$!3!********** ****R#!#<$\"3YM#[!\\<.46!#=7$$!33+++gOP&H#F*$\"3:!*3\"fz:1A\"F-7$$!3x* ****4%)QV?#F*$\"349=_a(*zE8F-7$$!34+++m1'>5#F*$\"3i&=GoA*Gd9F-7$$!3#** ****>'Q!*)*>F*$\"3oi(Rq\"yg,;F-7$$!3)*******ooL'*=F*$\"3!=$oss(G%f>F-7$$!37+++@6y-\"[$o-J1BF-7$$!3)*******3lW*\\\"F*$\"3k>?xua5 JDF-7$$!3))*****z9P]R\"F*$\"3A\"zBI:;_y#F-7$$!3.+++EL2.8F*$\"3h%4/3C'3 IIF-7$$!3!******R%Qa*>\"F*$\"3h(QWY00;L$F-7$$!37+++g#*e&4\"F*$\"31o*)y HYbkOF-7$$!3C******fv4a**F-$\"3d2?jA%fo,%F-7$$!3_,++!prV/*F-$\"3M^gVyb /mVF-7$$!3M******R'>E'zF-$\"3dbVfo>)4#[F-7$$!3C+++gPBYqF-$\"3p%eF-$\"3AL7]s#)RN$)F-7$$!3M******H4_U5F-$\"3y=/GIV&* )3*F-7$$!3+A-+++BpB!#?$\"3E$\\g@^9$y**F-7$$\"3i******4VeM5F-$\"3uqd2[n V*4\"F*7$$\"3K******4s\"e&>F-$\"37],)yfri>\"F*7$$\"3-,++!)Rx]HF-$\"3%> 00w)zX58F*7$$\"3U(*****z;myRF-$\"3S/%R(z=))R9F*7$$\"3o(*****f#\\U)\\F- $\"3k\\=wN&e)y:F*7$$\"3+)*****\\p?dfF-$\"3A(eRyq!3EF*7$$\"3&[*****R9@3!)F-$\"3'*H5?a'\\H3#F*7$$\"3H(*** ***zTY/*F-$\"3W\"e)z'ze/H#F*7$$\"3E%*****\\\"F*$\"3'H')fTZ \"R'*HF*7$$\"3))******Q([')H\"F*$\"3Q!ftDM$)oG$F*7$$\"3M******f%ztR\"F *$\"3C]zC6(*4)f$F*7$$\"3g*****\\PT2]\"F*$\"3y)R\\f@Lb&RF*7$$\"3.+++%= \"H+;F*$\"3h(49GG6LL%F*7$$\"3&******>C&4- +++6j0.=F*$\"3[=Dd$4]!=_F*7$$\"3'******z/He*=F*$\"37)p]t\")=5o&F*7$$\" 3()*****R2d@+#F*$\"37;=K%QlBE'F*7$$\"3a*****z7es4#F*$\"3&Q1-(f:bKoF*7$ $\"3-+++(ob')>#F*$\"3_TK-(Q'y(\\(F*7$$\"3%)******)31dH#F*$\"3iDmb_>3&> )F*7$$\"3!**************R#F*$\"3V$3?>Tuo,*F*-%&COLORG6&%$RGBG$\"#$*!\" #$\"\"!Fa[lF`[l-%*THICKNESSG6#\"\"#-F$6%7%7$F`[l$\"3++++++++DF*7$$\"\" \"Fa[lFj[l7$F]\\lF`[l-%'COLOURG6&F\\[lFa[lFa[lFa[l-%*LINESTYLEG6#\"\"$ -F$6&7$7$F`[lF]\\lF\\\\lF`\\l-%'SYMBOLG6$%'CIRCLEG\"#:-%&STYLEG6#%&POI NTG-F$6&Fi\\l-Fjz6&F\\[lF`[lF]\\lF`[l-F\\]l6$F^]l\"#5F`]l-F$6&Fi\\lFf] l-F\\]l6$%(DIAMONDGFj]lF`]l-F$6&Fi\\lFf]l-F\\]l6$%&CROSSGFj]lF`]l-%%TE XTG6%7$$\"$R#F_[l$!#>F_[lQ\"x6\"-%%FONTG6$%*HELVETICAG\"\"(-Ff^l6%7$$F *F_[l$\"$R&F_[lQ\"yF^_lF__l-Ff^l6%7$$F_[l!\"\"$\"$0\"F_[lQ\"1F^_lF__l- Ff^l6&7$$\"$%GF_[l$!#*)F_[lQ\"2F^_l-Fjz6&F\\[lF`[l$\"\"%F_`lF`[lF__l-F f^l6%7$F^`l$\"#DF_`lQ\"aF^_l-F`_l6%%&TIMESG%'ITALICG\"\")-F`_l6$Fb_lFi al-%(SCALINGG6#%.UNCONSTRAINEDG-%+AXESLABELSG6%%!GFcbl-F`_l6#%(DEFAULT G-%*AXESTICKSG6$7#/F^\\l%\"1G7#/FbalFcbl-%%VIEWG6$;$!$R#F_[lFi^l;$!\"$ F_`lFh_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cur ve 8" "Curve 9" "Curve 10" "Curve 11" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 11 "The points " }{XPPEDIT 18 0 "``(0,1)" "6#-%!G6$\"\"!\" \"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "``(1,a)" "6#-%!G6$\"\"\"%\" aG" }{TEXT -1 30 " are always on the graph of " }{XPPEDIT 18 0 "y=a^ x" "6#/%\"yG)%\"aG%\"xG" }{TEXT -1 7 ", with " }{XPPEDIT 18 0 "``(1,a) " "6#-%!G6$\"\"\"%\"aG" }{TEXT -1 28 " above and to the right of " } {XPPEDIT 18 0 "``(0,1)" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 7 " when " } {TEXT 395 1 "a" }{TEXT -1 7 " > 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "0H ]\\\"e\"F*7$$!3;p;a=3SDUFX$\"3wR3P+i!GZ\"F*7$$!32NL$eF/9I$FX$\"3;&))*G ?RC`8F*7$$!3*QLLL+e'=DFX$\"3kxF3'*\\ef7F*7$$!3v,](o+Q\"3;FX$\"33==],?w e6F*7$$!3RkLL$GN<-)!#>$\"3S?z%z8ri2\"F*7$$\"3m)))*\\7Vs4#)!#?$\"3!yCw% )Gd]#**FX7$$\"3M$**\\PH@8C*Feq$\"3q1g`8n3)=*FX7$$\"3Xmm\"z>2F!=FX$\"3' oz$HlwSx%)FX7$$\"3Dl;HsL^4EFX$\"3SOolPSItyFX7$$\"3GIL3FGwzMFX$\"37lj%* yW')psFX7$$\"3?I$e9V2PQ%FX$\"3#pu*Hp;)>p'FX7$$\"3M(*\\(o'Qfq^FX$\"3f60 3\"fikA'FX7$$\"3hLLeuFX?gFX$\"3?()e!e([**fdFX7$$\"3M%****\\=S%)*oFX$\" 3@^lwWwv9`FX7$$\"3=%**\\idztv(FX$\"3P.)>0-.D\"\\FX7$$\"3)y*\\7.wW)e)FX $\"31Zq*)[XI_XFX7$$\"3Z&**\\P&p?6&*FX$\"3)f>JQ0BK=%FX7$$\"3*emmrsMS.\" F*$\"3#HLrI@#=xQFX7$$\"3g***\\P:jD7\"F*$\"3MNv%)*)=4vNFX7$$\"3vKe9mIy- 7F*$\"3mBSY!GJY[!H\"F*$\"3!Gt(G*QS_1$FX7$$\"3yl\"zf %\\+t8F*$\"3Ein@B_,UGFX7$$\"3k*\\iXYi#f9F*$\"3uDuEf%Qgi#FX7$$\"39mmTb \\fV:F*$\"3FRt,$[e2V#FX7$$\"3K*\\7yD$)=j\"F*$\"3JU&\\'\\$f=C#FX7$$\"3H LLL'Q:pr\"F*$\"3[#)Quu\"=Q2#FX7$$\"3wK$3#>I(Q!=F*$\"3**3#p(y5*\\\">FX7 $$\"3Wm\"z*)f5,*=F*$\"3;r3dI^\\pF*$\"3eL01IecX;F X7$$\"3/m;/Me 1 occurs most frequently. In t his case both of the functions f, where " }{XPPEDIT 18 0 "f(x)=a^x" " 6#/-%\"fG6#%\"xG)%\"aGF'" }{TEXT -1 30 ", and the inverse function f " }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 11 ", given by " }}{PARA 0 "" 0 "" {TEXT -1 1 "f" }{XPPEDIT 18 0 "``^(-1);" "6#)% !G,$\"\"\"!\"\"" }{TEXT -1 1 "(" }{TEXT 397 1 "x" }{TEXT -1 1 ")" } {XPPEDIT 18 0 "`` = log[a]*x" "6#/%!G*&&%$logG6#%\"aG\"\"\"%\"xGF*" } {TEXT -1 47 " are increasing functions. While the graph of " } {XPPEDIT 18 0 "y=a^x" "6#/%\"yG)%\"aG%\"xG" }{TEXT -1 27 " passes thro ugh the points " }{XPPEDIT 18 0 "``(0,1)" "6#-%!G6$\"\"!\"\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "``(1,a)" "6#-%!G6$\"\"\"%\"aG" } {TEXT -1 16 ", the graph of " }{XPPEDIT 18 0 "y=log[a]*x" "6#/%\"yG*& &%$logG6#%\"aG\"\"\"%\"xGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "passes through the points " }{XPPEDIT 18 0 "``(1,0)" "6#-%!G6$ \"\"\"\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(a,1)" "6#-%!G6$%\" aG\"\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "log[a]*1=0" "6#/*&&%$logG6#%\"aG\"\"\"F)F)\"\"!" } {TEXT -1 15 " because " }{XPPEDIT 18 0 "a^0=1" "6#/*$%\"aG\"\"! \"\"\"" }{TEXT -1 22 ", " }{XPPEDIT 18 0 "log[a]*a =1" "6#/*&&%$logG6#%\"aG\"\"\"F(F)F)" }{TEXT -1 15 " because " } {XPPEDIT 18 0 "a^``(1)=a" "6#/)%\"aG-%!G6#\"\"\"F%" }{TEXT -1 2 ". 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This coincides with the range of the corresponding exponential f unction." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 5 "range" } {TEXT -1 6 " of " }{XPPEDIT 18 0 "log[a]" "6#&%$logG6#%\"aG" }{TEXT -1 32 " is the set of all real numbers " }{XPPEDIT 18 0 "``(-infinity, infinity)" "6#-%!G6$,$%)infinityG!\"\"F'" }{TEXT -1 75 ". This coincid es with the domain of the corresponding exponential function." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 " Properties of logarithms " }}{PARA 0 "" 0 "" {TEXT -1 99 "The standar d rules for exponents or powers lead to the following 3 basic properti es of logarithms: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "log[a](u*v) = log[a]*u+log[a]*v" "6#/ -&%$logG6#%\"aG6#*&%\"uG\"\"\"%\"vGF,,&*&&F&6#F(F,F+F,F,*&&F&6#F(F,F-F ,F," }{TEXT -1 15 " ------- (i), " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "log[a](u/v) = log[a]*u-log[a]*v" "6#/-&%$logG6#%\"aG 6#*&%\"uG\"\"\"%\"vG!\"\",&*&&F&6#F(F,F+F,F,*&&F&6#F(F,F-F,F." }{TEXT -1 16 " ------- (ii), " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[a](u^p) = p*log[a]*u" "6#/-&%$logG6#%\"aG6#)%\"uG%\"pG*(F,\" \"\"&F&6#F(F.F+F." }{TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The properties (i), (ii) \+ and (iii) could be called the " }{TEXT 262 45 "multiplication, divisio n and power properties" }{TEXT -1 29 " of logarithms respectively. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "log[a](a^x) = x" "6#/-&%$logG6#%\"aG6#)F(%\"xGF+" } {TEXT -1 16 " ------- (iv), " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a^(log[a]*x) = x" "6#/)%\"aG*&&%$logG6#F%\"\"\"%\"xGF*F +" }{TEXT -1 15 " ------- (v). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "The properties (iv) and (v) are the " } {TEXT 262 27 "inverse function identities" }{TEXT -1 42 " for exponent ial and logarithm functions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[b]*x = log[a]*x/( log[a]*b);" "6#/*&&%$logG6#%\"bG\"\"\"%\"xGF)*(&F&6#%\"aGF)F*F)*&&F&6# F.F)F(F)!\"\"" }{TEXT -1 15 " ------- (vi). " }}{PARA 0 "" 0 "" {TEXT -1 24 "The formula (vi) is the " }{TEXT 262 22 "change of base formula " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Proofs and explanations of the formulas (i) to (vi)" }} {PARA 0 "" 0 "" {TEXT -1 57 "The formula (i) follows from the sum rule for exponents: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a ^x*`.`*a^y = a^(x+y);" "6#/*()%\"aG%\"xG\"\"\"%\".GF()F&%\"yGF()F&,&F' F(F+F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "x=log[a]*u" "6#/%\"xG*&&%$logG6#%\"aG\"\"\"%\"uGF*" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "y=log[a]*v" "6#/%\"yG*&&%$logG6#% \"aG\"\"\"%\"vGF*" }{TEXT -1 11 ", so that " }{XPPEDIT 18 0 "u=a^x" " 6#/%\"uG)%\"aG%\"xG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "v = a^y;" "6 #/%\"vG)%\"aG%\"yG" }{TEXT -1 8 ". Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u*`.`*v=a^x*`.`*a^y" "6#/*(%\"uG\"\"\"%\".GF& %\"vGF&*()%\"aG%\"xGF&F'F&)F+%\"yGF&" }{XPPEDIT 18 0 "`` = a^(x+y)" "6 #/%!G)%\"aG,&%\"xG\"\"\"%\"yGF)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x+y=log[a](u*`.`*v)" "6#/,&%\"xG\"\"\"%\"yGF&-&%$logG6#%\"aG6#*(%\" uGF&%\".GF&%\"vGF&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "tha t is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[a](u*v) = log[a]*u+log[a]*v" "6#/-&%$logG6#%\"aG6#*&%\"uG\"\"\"%\"vGF,,&*&&F& 6#F(F,F+F,F,*&&F&6#F(F,F-F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 79 "The formula (ii) can be proved in a similar way using the rules for exponents " }{XPPEDIT 18 0 "a^x/a^y = a^(x-y)" "6#/*&)%\"a G%\"xG\"\"\")F&%\"yG!\"\")F&,&F'F(F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "To prove the formula (iii) let " }{XPPEDIT 18 0 "x= log[a]*u" "6#/%\"xG*&&%$logG6#%\"aG\"\"\"%\"uGF*" }{TEXT -1 11 ", so t hat " }{XPPEDIT 18 0 "u=a^x" "6#/%\"uG)%\"aG%\"xG" }{TEXT -1 8 ". Th en " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u^p=(a^x)^p" " 6#/)%\"uG%\"pG))%\"aG%\"xGF&" }{XPPEDIT 18 0 "``=a^(x*`.`*p)" "6#/%!G) %\"aG*(%\"xG\"\"\"%\".GF)%\"pGF)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*`.`*p=log[a](u^p)" "6#/*(%\"xG\"\"\"%\".GF&%\"pGF&-&%$logG6#%\"aG 6#)%\"uGF(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[a](u^p)=p*log[a ]*u" "6#/-&%$logG6#%\"aG6#)%\"uG%\"pG*(F,\"\"\"&F&6#F(F.F+F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The two formulas (iv) and (v) come from the fact that f where " } {XPPEDIT 18 0 "f(x)=a^x" "6#/-%\"fG6#%\"xG)%\"aGF'" }{TEXT -1 27 " has the inverse function f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\" " }{TEXT -1 9 ", where f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\" \"" }{TEXT -1 1 "(" }{TEXT 465 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "` `=log[a]*x" "6#/%!G*&&%$logG6#%\"aG\"\"\"%\"xGF*" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 23 "The general statement f" }{XPPEDIT 18 0 " ``^(-1)" "6#)%!G,$\"\"\"!\"\"" }{XPPEDIT 18 0 "``(f(x)) = x;" "6#/-%!G 6#-%\"fG6#%\"xGF*" }{TEXT -1 17 " for any number " }{TEXT 467 1 "x" } {TEXT -1 30 " in the domain of f becomes " }{XPPEDIT 18 0 "log[a](a^ x) = x" "6#/-&%$logG6#%\"aG6#)F(%\"xGF+" }{TEXT -1 24 " for every rea l number " }{TEXT 466 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "The general statement f(f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$ \"\"\"!\"\"" }{TEXT -1 1 "(" }{TEXT 469 1 "x" }{TEXT -1 2 "))" } {XPPEDIT 18 0 "``=x" "6#/%!G%\"xG" }{TEXT -1 17 " for any number " } {TEXT 470 1 "x" }{TEXT -1 20 " in the domain of f" }{XPPEDIT 18 0 "`` ^(-1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 10 " becomes " }{XPPEDIT 18 0 "a^(log[a]*x) = x" "6#/)%\"aG*&&%$logG6#F%\"\"\"%\"xGF*F+" }{TEXT -1 31 " for any positive real number " }{TEXT 468 1 "x" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 88 "However, (iv) and (v) can be checked directly. Equation (iv) follows by substituting " }{XPPEDIT 18 0 "y =a^x " "6#/%\"yG)%\"aG%\"xG" }{TEXT -1 6 " for " }{TEXT 472 1 "y" } {TEXT -1 29 " in the equivalent equation " }{XPPEDIT 18 0 "log[a]*y=x " "6#/*&&%$logG6#%\"aG\"\"\"%\"yGF)%\"xG" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 44 "while equation (v) follows by substituting " } {XPPEDIT 18 0 "y = log[a]*x;" "6#/%\"yG*&&%$logG6#%\"aG\"\"\"%\"xGF*" }{TEXT -1 6 " for " }{TEXT 471 1 "y" }{TEXT -1 29 " in the equivalent equation " }{XPPEDIT 18 0 "a^y=x" "6#/)%\"aG%\"yG%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "To prove the formula (vi) let " }{XPPEDIT 18 0 "y=log[b]*x" "6#/%\"yG*& &%$logG6#%\"bG\"\"\"%\"xGF*" }{TEXT -1 10 ", so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b^y = x;" "6#/)%\"bG%\"yG%\"xG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Then applying the function " }{XPPEDIT 18 0 "log[a]" "6#& %$logG6#%\"aG" }{TEXT -1 39 " to both sides of this equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[a](b^y) = log[a ]*x;" "6#/-&%$logG6#%\"aG6#)%\"bG%\"yG*&&F&6#F(\"\"\"%\"xGF0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y*log[a]*b = log[a]*x;" "6#/*(%\"yG\"\" \"&%$logG6#%\"aGF&%\"bGF&*&&F(6#F*F&%\"xGF&" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=log[a]*x/(log[a]*b)" "6#/%\"yG*(&%$logG6#% \"aG\"\"\"%\"xGF**&&F'6#F)F*%\"bGF*!\"\"" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "log[b]*x = log[a]*x/(log[a]*b)" "6#/*&&%$logG6#%\"bG\" \"\"%\"xGF)*(&F&6#%\"aGF)F*F)*&&F&6#F.F)F(F)!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "In words \+ \"the logarithm to a new base " }{TEXT 473 1 "b" }{TEXT -1 22 " of a p ositive number " }{TEXT 475 1 "x" }{TEXT -1 43 " is equal to the logar ithm to the old base " }{TEXT 478 1 "a" }{TEXT -1 15 " of the number \+ " }{TEXT 474 1 "x" }{TEXT -1 42 " divided by the logarithm to the old \+ base " }{TEXT 476 1 "a" }{TEXT -1 17 " of the new base " }{TEXT 477 1 "b" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 33 "Logarithms to ba se 10 are called " }{TEXT 262 17 "common logarithms" }{TEXT -1 169 ". \+ When the base of a logarithm is omitted it is usually understood to be 10. Common logarithms can be evaluated on a standard scientific calcu lator using the \"log\" key. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 41 "We can determine approximate values for " }{XPPEDIT 18 0 "log[10]*n" "6#*&&%$logG6#\"#5\"\"\"%\"nGF(" } {TEXT -1 7 " for " }{XPPEDIT 18 0 "n = 1,2,3,4,5,6,7,8,9,10;" "6,/% \"nG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5" }{TEXT -1 37 " using only approximate values for " }{XPPEDIT 18 0 "log[10]*2" "6#* &&%$logG6#\"#5\"\"\"\"\"#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "log[10] *3;" "6#*&&%$logG6#\"#5\"\"\"\"\"$F(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "log[10]*7;" "6#*&&%$logG6#\"#5\"\"\"\"\"(F(" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[10]*2" "6#*&&%$lo gG6#\"#5\"\"\"\"\"#F(" }{TEXT -1 1 " " }{TEXT 413 1 "~" }{TEXT -1 20 " 0.301030 and " }{XPPEDIT 18 0 "log[10]*3;" "6#*&&%$logG6#\"#5\" \"\"\"\"$F(" }{TEXT -1 1 " " }{TEXT 414 1 "~" }{TEXT -1 14 " 0.477121( 3). " }}{PARA 0 "" 0 "" {TEXT -1 33 "Adding these two logarithms gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[10]*6;" "6#*& &%$logG6#\"#5\"\"\"\"\"'F(" }{TEXT -1 1 " " }{TEXT 415 1 "~" }{TEXT -1 11 " 0.778151. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Also" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[10]*5 = log[10](10/2);" "6#/ *&&%$logG6#\"#5\"\"\"\"\"&F)-&F&6#F(6#*&F(F)\"\"#!\"\"" }{XPPEDIT 18 0 "``=log[10]*10-log[10]*2" "6#/%!G,&*&&%$logG6#\"#5\"\"\"F*F+F+*&&F(6 #F*F+\"\"#F+!\"\"" }{TEXT -1 1 " " }{TEXT 416 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "1-0" "6#,&\"\"\"F$\"\"!!\"\"" }{TEXT -1 20 ".301030 = 0 .698970. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[10]*7" "6#*&&%$logG6#\"#5\"\"\"\"\"(F(" } {TEXT -1 1 " " }{TEXT 417 1 "~" }{TEXT -1 11 " 0.845098. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[10]*8 = log[10](2^3);" "6#/*&&%$logG6#\"#5\"\"\"\"\")F)-&F&6#F( 6#*$\"\"#\"\"$" }{XPPEDIT 18 0 "``=3*log[10]*2" "6#/%!G*(\"\"$\"\"\"&% $logG6#\"#5F'\"\"#F'" }{TEXT -1 1 " " }{TEXT 418 1 "~" }{TEXT -1 11 " \+ 0.903090. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[10]*9 = log[10](3^2);" "6#/*&&%$logG6#\"# 5\"\"\"\"\"*F)-&F&6#F(6#*$\"\"$\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 " ``=2*log[10]*3" "6#/%!G*(\"\"#\"\"\"&%$logG6#\"#5F'\"\"$F'" }{TEXT -1 1 " " }{TEXT 419 1 "~" }{TEXT -1 11 " 0.954243. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "This gives the following table of approximate values which can be used to help in plotting a g raph of " }{XPPEDIT 18 0 "y = log[10]*x" "6#/%\"yG*&&%$logG6#\"#5\"\" \"%\"xGF*" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "1<=x" "6#1\"\"\"%\"xG " }{XPPEDIT 18 0 "``<=10" "6#1%!G\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " matrix([[x,`|`,1,2,3,4,5,6,7,8,9,10],[log[10]*x,`|`,0,0*.301030, 0*.47 7121, 0*.602060, 0*.698970, 0*.778151, 0*.845098, 0*.903090, 0*.954243 , 1]])" "6#-%'matrixG6#7$7.%\"xG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"' \"\"(\"\")\"\"*\"#57.*&&%$logG6#F3F*F(F*F)\"\"!*&F9F*-%&FloatG6$\"'I5I !\"'F**&F9F*-F<6$\"'@rZF?F**&F9F*-F<6$\"'g?gF?F**&F9F*-F<6$\"'q*)pF?F* *&F9F*-F<6$\"'^\"y(F?F**&F9F*-F<6$\"')4X)F?F**&F9F*-F<6$\"'!4.*F?F**&F 9F*-F<6$\"'VU&*F?F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 272 277 277 {PLOTDATA 2 "6J-%'CURVESG6%7gn7$$\"3-++++++ +5!#>$!3c**************>!#<7$$\"39+vV[$>#4=F*$!3#e2,kx3Du\"F-7$$\"3V+] (opQ%=EF*$!3)eR&Q*ed>e\"F-7$$\"3Q+DJX!ewU$F*$!3YA]E6D+l9F-7$$\"3n++v$R xoB%F*$!3u<)yL5aHP\"F-7$$\"3E,]i!4;`&eF*$!3`%45_l\\CB\"F-7$$\"3%=++vya PZ(F*$!3(Rat`:hk7\"F-7$$\"3;+]7=K1r5!#=$!3OH(*RZ*[=q*FO7$$\"3G++]d4v%R \"FO$!35\"\\KxJL]b)FO7$$\"3B++DOk7U?FO$!3Qu#4]st\"**oFO7$$\"3Y+++:>]*o #FO$!3JCI_@9G.dFO7$$\"3i+]7+y0;QFO$!3'QjL&e0&Q=%FO7$$\"3y++D&o8E%\\FO$ !3'>.^nKL/1$FO7$$\"3q++];NZwuFO$!3-fwE2?.j7FO7$$\"3C++bTHr-5F-$\"3kg\\ k9-iw6!#?7$$\"3/+]A\\icc7F-$\"32VYAf!R&=**F*7$$\"3/+]#4w?>\\\"F-$\"39- R[uvXPm#QXR#FO7$$\"3A+]sSuk()>F-$\"39f&3% QNR$)HFO7$$\"3;+]A!*)p)QAF-$\"3oBM0d&)G+NFO7$$\"3M++qeIG(\\#F-$\"3gDTr 4xnuRFO7$$\"3.++:=N*[s#F-$\"3KLvHe`\\`VFO7$$\"3K++5')*G6)HF-$\"3HbX!Hn 2Qu%FO7$$\"37++]\"e;%QKF-$\"3sZODXrK.^FO7$$\"3S++!R&3O'[$F-$\"3ong;lLs BaFO7$$\"3)***\\s1!=:r$F-$\"3i\"e!3>d^&p&FO7$$\"39++5R;DzRF-$\"3)o\"GS QS,)*fFO7$$\"3*)****RWr01UF-$\"3cM$*3)o^(QiFO7$$\"3w**\\Kle))pWF-$\"3m -,RRV'H]'FO7$$\"33++S()*=Mq%F-$\"3#omI0oOTs'FO7$$\"3q+]KN=kf\\F-$\"3-F 2%o9.X&pFO7$$\"31,](zdEO?&F-$\"3[fM>K71jrFO7$$\"35,+Dh!*>eaF-$\"3_Rc+/ P\\qtFO7$$\"3O+]#)pg(>p&F-$\"3)\\tvblIEb(FO7$$\"3()***\\d:OT%fF-$\"3pH ip([()3u(FO7$$\"3i+]s;'fg?'F-$\"3UL4$>Vf\"GzFO7$$\"3X+]ZfZ1MkF-$\"3G,I .yU&[3)FO7$$\"3_++0glJ!o'F-$\"3y&z&oG/(zC)FO7$$\"3Y++!ew=Z$pF-$\"3Fggh J&)G5%)FO7$$\"3')***\\=p,O=(F-$\"3#R:>1WAMc)FO7$$\"3>+]7?(3WU(F-$\"3ea (f:s=mq)FO7$$\"3%3+]&oQy\"p(F-$\"3A,R'\\sq-'))FO7$$\"3T****RJB.KzF-$\" 3QY1XdZ%Q**)FO7$$\"3],+]X)[&)=)F-$\"3(\\3(yR%p?8*FO7$$\"3C**\\7L4*4U)F -$\"3W1z\\(*>j`#*FO7$$\"3k++!4`6^n)F-$\"3C%)>sp1v#Q*FO7$$\"3B**\\K4#>U \"*)F-$\"3ZXS0'4L3]*FO7$$\"39+]-Hc:k\"*F-$\"3hG)o#fX#4i*FO7$$\"3e**** \\jT^3%*F-$\"3m%*QHI/@N(*FO7$$\"3w+]7.\\Lk'*F-$\"3yW%fOr><&)*FO7$$\"3w ++S!o?2\"**F-$\"3I$=gLO_5'**FO7$$\"3-+]Rs&oi,\"!#;$\"3'[TVa\\3q+\"F-7$ $\"3C+D(pVc7/\"Fg[l$\"3EM/?*pdv,\"F-7$$\"3A++Q*o\"F-7$F\\ilFg^mF_dl-%%TEXTG6$7$$\"%p7Fc^l$!\"& Fc^lQ\"x6\"-F[_m6$7$$!#DFc^l$\"$H\"Fc^lQ\"yFc_m-%%FONTG6$%*HELVETICAGF j`l-%*AXESTICKSG6$7//Fe^lFe^l/F__l%\"1G/Fi^l%\"2G/Ff_l%\"3G/F[`l%\"4G/ F``l%\"5G/Fe`l%\"6G/Fj`l%\"7G/F_al%\"8G/Fdal%\"9G/Fial%#10G/Fjgl%#11G/ Fahl%#12G70/$Fc^lFbdl%%-0.2G/$FbdlFbdl%%-0.1G/$F__lFbdl%$0.1G/$Fi^lFbd l%$0.2G/$Ff_lFbdl%$0.3G/$F[`lFbdl%$0.4G/$F``lFbdl%$0.5G/Fadl%$0.6G/$Fj `lFbdl%$0.7G/$F_alFbdl%$0.8G/$FdalFbdl%$0.9G/F__l%$1.0G/$FjglFbdl%$1.1 G/$FahlFbdl%$1.2G-%(SCALINGG6#%.UNCONSTRAINEDG-%+AXESLABELSG6%%!GF]em- F]`m6#%(DEFAULTG-%%VIEWG6$;Fg_mF^_m;Fg_mFi_m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "C urve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 37 "We determine app roximate values for " }{XPPEDIT 18 0 "log[2]*n;" "6#*&&%$logG6#\"\"# \"\"\"%\"nGF(" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "n = 1,2,3,4,5,6,7 ,8,9,10;" "6,/%\"nG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5 " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "x=log[2]*3" "6#/%\"xG*&&%$logG6#\"\"#\"\"\"\"\"$F*" }{TEXT -1 8 ", th en " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2^x = 3;" "6# /)\"\"#%\"xG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "App lying the function " }{XPPEDIT 18 0 "log[10]" "6#&%$logG6#\"#5" } {TEXT -1 39 " to both sides of this equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[10]*2^x = log[10]*3;" "6#/*&&% $logG6#\"#5\"\"\")\"\"#%\"xGF)*&&F&6#F(F)\"\"$F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 46 "Using the power property of logarithms gi ves: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*log[10]*2= log[10]*3" "6#/*(%\"xG\"\"\"&%$logG6#\"#5F&\"\"#F&*&&F(6#F*F&\"\"$F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=log[10]*3/(log[10]*2)" "6#/%\" xG*(&%$logG6#\"#5\"\"\"\"\"$F**&&F'6#F)F*\"\"#F*!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 421 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "0*.477121/(0*.301030) " "6#*(\"\"!\"\"\"-%&FloatG6$\"'@rZ!\"'F%*&F$F%-F'6$\"'I5IF*F%!\"\"" } {TEXT -1 1 " " }{TEXT 420 1 "~" }{TEXT -1 10 " 1.58496. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "The steps used he re follow those used to prove the change of base formula and, of cours e, the change of base formula gives immediately that: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[2]*3 = log[10]*3/(log[10]*2) " "6#/*&&%$logG6#\"\"#\"\"\"\"\"$F)*(&F&6#\"#5F)F*F)*&&F&6#F.F)F(F)!\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "We also have " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[2]*5 = log[10]*5/ (log[10]*2)" "6#/*&&%$logG6#\"\"#\"\"\"\"\"&F)*(&F&6#\"#5F)F*F)*&&F&6# F.F)F(F)!\"\"" }{TEXT -1 1 " " }{TEXT 422 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "0*.698970/(0*.301030)" "6#*(\"\"!\"\"\"-%&FloatG6$\"'q* )p!\"'F%*&F$F%-F'6$\"'I5IF*F%!\"\"" }{TEXT -1 1 " " }{TEXT 423 1 "~" } {TEXT -1 20 " 2.32193 " }{XPPEDIT 18 0 "log[2]*6 = log[10]* 6/(log[10]*2)" "6#/*&&%$logG6#\"\"#\"\"\"\"\"'F)*(&F&6#\"#5F)F*F)*&&F& 6#F.F)F(F)!\"\"" }{TEXT -1 1 " " }{TEXT 424 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "0*.778151/(0*.301030)" "6#*(\"\"!\"\"\"-%&FloatG6$\"'^ \"y(!\"'F%*&F$F%-F'6$\"'I5IF*F%!\"\"" }{TEXT -1 1 " " }{TEXT 425 1 "~ " }{TEXT -1 9 " 2.58496 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[2]*9 = log[10]*9/(log[10]*2 );" "6#/*&&%$logG6#\"\"#\"\"\"\"\"*F)*(&F&6#\"#5F)F*F)*&&F&6#F.F)F(F)! \"\"" }{TEXT -1 1 " " }{TEXT 426 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "0*.954243/(0*.301030)" "6#*(\"\"!\"\"\"-%&FloatG6$\"'VU&*!\"'F%*&F$F% -F'6$\"'I5IF*F%!\"\"" }{TEXT -1 1 " " }{TEXT 427 1 "~" }{TEXT -1 21 " \+ 3.16993 " }{XPPEDIT 18 0 "log[2]*10 = log[10]*10/(log[10]* 2);" "6#/*&&%$logG6#\"\"#\"\"\"\"#5F)*(&F&6#F*F)F*F)*&&F&6#F*F)F(F)!\" \"" }{TEXT -1 1 " " }{TEXT 428 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1 /(0*.301030)" "6#*&\"\"\"F$*&\"\"!F$-%&FloatG6$\"'I5I!\"'F$!\"\"" } {TEXT -1 1 " " }{TEXT 429 1 "~" }{TEXT -1 9 " 3.32193 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "This gives the foll owing table of approximate values which can be used to help in plottin g a graph of " }{XPPEDIT 18 0 "y = log[2]*x;" "6#/%\"yG*&&%$logG6#\" \"#\"\"\"%\"xGF*" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "1<=x" "6#1\"\" \"%\"xG" }{XPPEDIT 18 0 "``<=10" "6#1%!G\"#5" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[x, `|`, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [log[2 ]*x, `|`, 0, 1, 1.58496, 2, 2.32193, 2.58496, 2.80735, 3, 3.16993, 3.3 2193]]);" "6#-%'matrixG6#7$7.%\"xG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\" \"'\"\"(\"\")\"\"*\"#57.*&&%$logG6#F+F*F(F*F)\"\"!F*-%&FloatG6$\"''\\e \"!\"&F+-F;6$\"'$>K#F>-F;6$\"''\\e#F>-F;6$\"'N2GF>F,-F;6$\"'$*pJF>-F;6 $\"'$>K$F>" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 249 236 236 {PLOTDATA 2 "6O-%'CU RVESG6%7gn7$$\"3-+++++++5!#>$!3OCZx*=cQk'!#<7$$\"39+vV[$>#4=F*$!3ANX\" *f))[)y&F-7$$\"3V+](opQ%=EF*$!3;L'*4h$\\^D&F-7$$\"3Q+DJX!ewU$F*$!3U==D ,Ijm[F-7$$\"3n++v$RxoB%F*$!3'Q)\\+4[&3c%F-7$$\"3E,]i!4;`&eF*$!33cm%GF-7$$\"3B++DOk7U?FO$!3#4qhG!f&=H#F- 7$$\"3Y+++:>]*o#FO$!3=Z1sy!*e%*=F-7$$\"3i+]7+y0;QFO$!3M1._r]%)*Q\"F-7$ $\"3y++D&o8E%\\FO$!3'\\Qm]%Rl;5F-7$$\"3q++];NZwuFO$!3M;Wo4:q&>%FO7$$\" 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-Fjal6$%&CROSSGFcalF^bl-F$6$7$7$Fd^l$!3++++++++DFO7$Fd^l$\"3#)******** *****z$F--F^^l6&F`^l$Fa`l!\"\"F]dlF]dl-F$6$7$7$F^_lFfcl7$F^_lFiclF[dl- F$6$7$7$Fa_lFfcl7$Fa_lFiclF[dl-F$6$7$7$Fc_lFfcl7$Fc_lFiclF[dl-F$6$7$7$ Fh_lFfcl7$Fh_lFiclF[dl-F$6$7$7$F[`lFfcl7$F[`lFiclF[dl-F$6$7$7$F``lFfcl 7$F``lFiclF[dl-F$6$7$7$Fe`lFfcl7$Fe`lFiclF[dl-F$6$7$7$Fj`lFfcl7$Fj`lFi clF[dl-F$6$7$7$F]alFfcl7$F]alFiclF[dl-F$6$7$7$FbalFfcl7$FbalFiclF[dl-F $6$7$7$$\"#6Fe^lFfcl7$FeglFiclF[dl-F$6$7$7$$\"#7Fe^lFfcl7$F\\hlFiclF[d l-F$6$7$7$$!3/+++++++5FO$!3A+++++++SFO7$$\"#8Fe^lFehlF[dl-F$6$7$7$Fchl $!35+++++++?FO7$FhhlF^ilF[dl-F$6$7$7$FchlFd^l7$FhhlFd^lF[dl-F$6$7$7$Fc hl$\"35+++++++?FO7$FhhlFjilF[dl-F$6$7$7$Fchl$\"3A+++++++SFO7$FhhlFajlF [dl-F$6$7$7$Fchl$\"3w**************fFO7$FhhlFhjlF[dl-F$6$7$7$Fchl$\"3U +++++++!)FO7$FhhlF_[mF[dl-F$6$7$7$FchlF^_l7$FhhlF^_lF[dl-F$6$7$7$Fchl$ \"3%**************>\"F-7$FhhlF[\\mF[dl-F$6$7$7$Fchl$\"3!************** R\"F-7$FhhlFb\\mF[dl-F$6$7$7$Fchl$\"33+++++++;F-7$FhhlFi\\mF[dl-F$6$7$ 7$Fchl$\"3/+++++++=F-7$FhhlF`]mF[dl-F$6$7$7$FchlFa_l7$FhhlFa_lF[dl-F$6 $7$7$Fchl$\"3;+++++++AF-7$FhhlF\\^mF[dl-F$6$7$7$Fchl$\"3!************* *R#F-7$FhhlFc^mF[dl-F$6$7$7$Fchl$\"33+++++++EF-7$FhhlFj^mF[dl-F$6$7$7$ Fchl$\"3#)*************z#F-7$FhhlFa_mF[dl-F$6$7$7$FchlFc_l7$FhhlFc_lF[ dl-F$6$7$7$Fchl$\"3;+++++++KF-7$FhhlF]`mF[dl-F$6$7$7$Fchl$\"3!******** ******R$F-7$FhhlFd`mF[dl-%%TEXTG6$7$$\"%p7Fc^l$!\")Fc^lQ\"x6\"-Fh`m6$7 $$!#DFc^l$\"$f$Fc^lQ\"yF`am-%%FONTG6$%*HELVETICAGFf`l-%*AXESTICKSG6$7/ /Fe^lFe^l/F__l%\"1G/Fi^l%\"2G/Fd_l%\"3G/Fi_l%\"4G/F\\`l%\"5G/Fa`l%\"6G /Ff`l%\"7G/F[al%\"8G/F^al%\"9G/Fcal%#10G/Ffgl%#11G/F]hl%#12G75/$Fc^lF^ dl%%-0.2G/Fe^l%\"0G/$Fi^lF^dl%$0.2G/$Fi_lF^dl%$0.4G/F]dl%$0.6G/$F[alF^ dl%$0.8G/F__l%$1.0G/$F]hlF^dl%$1.2G/$\"#9F^dl%$1.4G/$\"#;F^dl%$1.6G/$ \"#=F^dl%$1.8G/Fi^l%$2.0G/$\"#AF^dl%$2.2G/$\"#CF^dl%$2.4G/$\"#EF^dl%$2 .6G/$\"#GF^dl%$2.8G/Fd_l%$3.0G/$\"#KF^dl%$3.2G/$\"#MF^dl%$3.4G-%(SCALI NGG6#%.UNCONSTRAINEDG-%+AXESLABELSG6%%!GF_hm-Fjam6#%(DEFAULTG-%%VIEWG6 $;FdamF[am;FdamFfam" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Cur ve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 3 8" "Curve 39" "Curve 40" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT 430 8 "Question" }{TEXT -1 53 ": Without using a calculator f ind the value of (a) " }{XPPEDIT 18 0 "log[8]*2" "6#*&&%$logG6#\"\") \"\"\"\"\"#F(" }{TEXT -1 10 " (b) " }{XPPEDIT 18 0 "log[10]*sqrt( 10)" "6#*&&%$logG6#\"#5\"\"\"-%%sqrtG6#F'F(" }{TEXT -1 10 " (c) \+ " }{XPPEDIT 18 0 "log[4]*8" "6#*&&%$logG6#\"\"%\"\"\"\"\")F(" }{TEXT -1 9 " (d) " }{XPPEDIT 18 0 "log[8]*4" "6#*&&%$logG6#\"\")\"\"\"\" \"%F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "log[8]*2=1/3" "6#/*&&%$logG6#\"\")\"\"\" \"\"#F)*&F)F)\"\"$!\"\"" }{TEXT -1 13 " because " }{XPPEDIT 18 0 " 8^(1/3)=2" "6#/)\"\")*&\"\"\"F'\"\"$!\"\"\"\"#" }{TEXT -1 11 " (b ) " }{XPPEDIT 18 0 "log[10]*sqrt(10)=1/2" "6#/*&&%$logG6#\"#5\"\"\"-% %sqrtG6#F(F)*&F)F)\"\"#!\"\"" }{TEXT -1 14 " because " }{XPPEDIT 18 0 "sqrt(10)=10^(1/2)" "6#/-%%sqrtG6#\"#5)F'*&\"\"\"F*\"\"#!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(c) " }{XPPEDIT 18 0 "lo g[4]*8=3/2" "6#/*&&%$logG6#\"\"%\"\"\"\"\")F)*&\"\"$F)\"\"#!\"\"" } {TEXT -1 14 " because " }{XPPEDIT 18 0 "4^(3/2)=8" "6#/)\"\"%*&\" \"$\"\"\"\"\"#!\"\"\"\")" }{TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "In more detail, let " } {XPPEDIT 18 0 "u=log[4]*8" "6#/%\"uG*&&%$logG6#\"\"%\"\"\"\"\")F*" } {TEXT -1 11 ", so that " }{XPPEDIT 18 0 "8=4^u" "6#/\"\"))\"\"%%\"uG " }{TEXT -1 11 ", that is " }{XPPEDIT 18 0 "2^3=2^(2*u)" "6#/*$\"\"# \"\"$)F%*&F%\"\"\"%\"uGF)" }{TEXT -1 10 ". Hence " }{XPPEDIT 18 0 "2 *u=3" "6#/*&\"\"#\"\"\"%\"uGF&\"\"$" }{TEXT -1 11 ", so that " } {XPPEDIT 18 0 "u=3/2" "6#/%\"uG*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 " ( d) l" }{XPPEDIT 18 0 "og[8]*4=2/3" "6#/*&&%#ogG6#\"\")\"\"\"\"\"%F)*& \"\"#F)\"\"$!\"\"" }{TEXT -1 14 " because " }{XPPEDIT 18 0 "8^(2/ 3)=4" "6#/)\"\")*&\"\"#\"\"\"\"\"$!\"\"\"\"%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "In more \+ detail, let " }{XPPEDIT 18 0 "u=log[8]*4" "6#/%\"uG*&&%$logG6#\"\")\" \"\"\"\"%F*" }{TEXT -1 11 ", so that " }{XPPEDIT 18 0 "4=8^u" "6#/\" \"%)\"\")%\"uG" }{TEXT -1 12 ", that is " }{XPPEDIT 18 0 "2^2=2^(3*u )" "6#/*$\"\"#F%)F%*&\"\"$\"\"\"%\"uGF)" }{TEXT -1 10 ". Hence " } {XPPEDIT 18 0 "3*u=2" "6#/*&\"\"$\"\"\"%\"uGF&\"\"#" }{TEXT -1 11 ", s o that " }{XPPEDIT 18 0 "u=2/3" "6#/%\"uG*&\"\"#\"\"\"\"\"$!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Alternatively, the change of base formula gives " } {XPPEDIT 18 0 "log[8]*4=log[4]*4/(log[4]*8)" "6#/*&&%$logG6#\"\")\"\" \"\"\"%F)*(&F&6#F*F)F*F)*&&F&6#F*F)F(F)!\"\"" }{XPPEDIT 18 0 "``=1/(lo g[4]*8)" "6#/%!G*&\"\"\"F&*&&%$logG6#\"\"%F&\"\")F&!\"\"" }{XPPEDIT 18 0 "``=1/``(3/2)" "6#/%!G*&\"\"\"F&-F$6#*&\"\"$F&\"\"#!\"\"F," } {XPPEDIT 18 0 "``=2/3" "6#/%!G*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT 437 8 "Question" }{TEXT -1 111 " : Find approximate values for the following by making use of common lo garithms and the change of base formula. " }}{PARA 256 "" 0 "" {TEXT -1 7 " (a) " }{XPPEDIT 18 0 "log[3]*20;" "6#*&&%$logG6#\"\"$\"\"\"\" #?F(" }{TEXT -1 8 " (b) " }{XPPEDIT 18 0 "log[Pi]*100;" "6#*&&%$log G6#%#PiG\"\"\"\"$+\"F(" }{TEXT -1 7 " (c) " }{XPPEDIT 18 0 "log[5.3] *1700;" "6#*&&%$logG6#-%&FloatG6$\"#`!\"\"\"\"\"\"%+ 0 in order for " } {XPPEDIT 18 0 "log[a]" "6#&%$logG6#%\"aG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "a/sqrt(x)" "6#*&%\"aG\"\"\"-%%sqrtG6#%\"xG!\"\"" }{TEXT -1 33 " \+ to be defined. This means that " }{XPPEDIT 18 0 "sqrt(x^2)" "6#-%%sqr tG6#*$%\"xG\"\"#" }{TEXT -1 15 " simplifies to " }{TEXT 441 1 "x" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 7" }}{PARA 0 "" 0 "" {TEXT 444 8 "Question" } {TEXT -1 13 ": Given that " }{XPPEDIT 18 0 " log[a]*x=2" "6#/*&&%$logG 6#%\"aG\"\"\"%\"xGF)\"\"#" }{TEXT -1 26 ", find the value of (a) " } {XPPEDIT 18 0 "log[a](1/x)" "6#-&%$logG6#%\"aG6#*&\"\"\"F*%\"xG!\"\"" }{TEXT -1 11 " (b) " }{XPPEDIT 18 0 "log[1/a]*x" "6#*&&%$logG6#* &\"\"\"F(%\"aG!\"\"F(%\"xGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 258 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 " (a) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " log[a](1/x)= log[a](x^(-1))" "6#/-&%$logG6#%\"aG6#*&\"\"\"F+%\"xG!\"\"-&F&6#F(6#)F, ,$F+F-" }{XPPEDIT 18 0 "``=-log[a]*x" "6#/%!G,$*&&%$logG6#%\"aG\"\"\"% \"xGF+!\"\"" }{XPPEDIT 18 0 "``=-2" "6#/%!G,$\"\"#!\"\"" }{TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[1/a]*x = log[a]*x/log[a](1/a)" "6#/*&&%$l ogG6#*&\"\"\"F)%\"aG!\"\"F)%\"xGF)*(&F&6#F*F)F,F)-&F&6#F*6#*&F)F)F*F+F +" }{XPPEDIT 18 0 "`` = log[a]*x/(-1);" "6#/%!G*(&%$logG6#%\"aG\"\"\"% \"xGF*,$F*!\"\"F-" }{XPPEDIT 18 0 "``=-2" "6#/%!G,$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }}{PARA 0 "" 0 "" {TEXT 456 8 "Question" } {TEXT -1 9 ": Given " }{XPPEDIT 18 0 "f(x) = 3^(x-2);" "6#/-%\"fG6#% \"xG)\"\"$,&F'\"\"\"\"\"#!\"\"" }{TEXT -1 10 ", find f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 1 "(" }{TEXT 445 1 "x" }{TEXT -1 35 ") and sketch the graphs of f and f" }{XPPEDIT 18 0 "``^ (-1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 28 " with the same set of axes. \+ " }}{PARA 0 "" 0 "" {TEXT 258 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 40 "To find the inverse function of f let " } {XPPEDIT 18 0 "y = 3^(x-2);" "6#/%\"yG)\"\"$,&%\"xG\"\"\"\"\"#!\"\"" } {TEXT -1 16 ", and solve for " }{TEXT 446 1 "x" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=3^(x-2)" "6#/%\"yG) \"\"$,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "is equivalent to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "log[3]*y=x-2" "6#/*&&%$logG6#\"\"$\"\"\"%\"yGF),&%\"xGF )\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "which, in \+ turn is equivalent to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[3]*y+2=x" "6#/,&*&&%$logG6#\"\"$\"\"\"%\"yGF*F*\"\"#F*%\"xG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 2 " f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" } {TEXT -1 1 "(" }{TEXT 449 1 "y" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "`` = l og[3]*y+2;" "6#/%!G,&*&&%$logG6#\"\"$\"\"\"%\"yGF+F+\"\"#F+" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "Replacing the variable " } {TEXT 450 1 "y" }{TEXT -1 4 " by " }{TEXT 451 1 "x" }{TEXT -1 8 " give s: " }}{PARA 256 "" 0 "" {TEXT -1 2 " f" }{XPPEDIT 18 0 "``^(-1)" "6#) %!G,$\"\"\"!\"\"" }{TEXT -1 1 "(" }{TEXT 448 1 "x" }{TEXT -1 1 ")" } {XPPEDIT 18 0 "`` = log[3]*x+2;" "6#/%!G,&*&&%$logG6#\"\"$\"\"\"%\"xGF +F+\"\"#F+" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 447 9 "_________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph of " }{XPPEDIT 18 0 "y=3^( x-2)" "6#/%\"yG)\"\"$,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 42 " is obtain ed by translating the graph of " }{XPPEDIT 18 0 "y=3^x" "6#/%\"yG)\" \"$%\"xG" }{TEXT -1 32 " to the right by two units. The " }{TEXT 454 1 "y" }{TEXT -1 29 " intercept of the graph is at" }{XPPEDIT 18 0 "``( 0,1/9)" "6#-%!G6$\"\"!*&\"\"\"F(\"\"*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph of " }{XPPEDIT 18 0 "y=log[3]*x+2" " 6#/%\"yG,&*&&%$logG6#\"\"$\"\"\"%\"xGF+F+\"\"#F+" }{TEXT -1 42 " is ob tained by translating the graph of " }{XPPEDIT 18 0 "y=log[3]*x" "6#/ %\"yG*&&%$logG6#\"\"$\"\"\"%\"xGF*" }{TEXT -1 25 " upwards by 2 units. The " }{TEXT 455 1 "x" }{TEXT -1 24 " intercept is the point " } {XPPEDIT 18 0 "``(1/9,0)" "6#-%!G6$*&\"\"\"F'\"\"*!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "In the following picture the \+ graph of " }{XPPEDIT 18 0 "y=3^(x-2)" "6#/%\"yG)\"\"$,&%\"xG\"\"\"\" \"#!\"\"" }{TEXT -1 15 " is plotted in " }{TEXT 368 3 "red" }{TEXT -1 22 ", while the graph of " }{XPPEDIT 18 0 "y=log[3]*x+2" "6#/%\"yG,&* &&%$logG6#\"\"$\"\"\"%\"xGF+F+\"\"#F+" }{TEXT -1 15 " is plotted in " }{TEXT 260 4 "blue" }{TEXT -1 2 ". 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The oth er point of intersection occurs where " }{TEXT 453 1 "x" }{TEXT -1 1 " " }{TEXT 452 1 "~" }{TEXT -1 11 " 0.127869. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 9" }} {PARA 0 "" 0 "" {TEXT 464 8 "Question" }{TEXT -1 9 ": Given " } {XPPEDIT 18 0 "f(x) = log[5](x+1);" "6#/-%\"fG6#%\"xG-&%$logG6#\"\"&6# ,&F'\"\"\"F/F/" }{TEXT -1 10 ", find f" }{XPPEDIT 18 0 "``^(-1)" "6# )%!G,$\"\"\"!\"\"" }{TEXT -1 1 "(" }{TEXT 457 1 "x" }{TEXT -1 35 ") a nd sketch the graphs of f and f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\" \"\"!\"\"" }{TEXT -1 28 " with the same set of axes. " }}{PARA 0 "" 0 "" {TEXT 258 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 40 "To find the inverse function of f let " }{XPPEDIT 18 0 "y = log[ 5](x+1);" "6#/%\"yG-&%$logG6#\"\"&6#,&%\"xG\"\"\"F-F-" }{TEXT -1 16 ", and solve for " }{TEXT 458 1 "x" }{TEXT -1 2 ". " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "y=log[5](x+1)" "6#/%\"yG-&%$logG6#\" \"&6#,&%\"xG\"\"\"F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "is equivalent to " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "5^y=x+1" "6#/)\"\"&%\"yG,&%\"xG\"\"\"F)F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Thus " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=5^y-1" "6#/%\"xG,&)\"\"&%\"yG\"\"\"F)!\"\"" }{TEXT -1 4 " = f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 1 "(" }{TEXT 463 1 "y" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 6 "H ence " }}{PARA 256 "" 0 "" {TEXT -1 2 " f" }{XPPEDIT 18 0 "``^(-1)" "6 #)%!G,$\"\"\"!\"\"" }{TEXT -1 1 "(" }{TEXT 460 1 "x" }{TEXT -1 1 ")" } {XPPEDIT 18 0 "`` = 5^x-1;" "6#/%!G,&)\"\"&%\"xG\"\"\"F)!\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 459 8 "________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph of " }{XPPEDIT 18 0 "y = log[5](x+1);" "6#/%\" yG-&%$logG6#\"\"&6#,&%\"xG\"\"\"F-F-" }{TEXT -1 42 " is obtained by tr anslating the graph of " }{XPPEDIT 18 0 "y = log[5]*x;" "6#/%\"yG*&&% $logG6#\"\"&\"\"\"%\"xGF*" }{TEXT -1 25 " to the left by one unit." }} {PARA 0 "" 0 "" {TEXT -1 10 "The line " }{XPPEDIT 18 0 "x=-1" "6#/%\" xG,$\"\"\"!\"\"" }{TEXT -1 40 " is a vertical asymptote. The graph of \+ " }{XPPEDIT 18 0 "y = 5^x-1;" "6#/%\"yG,&)\"\"&%\"xG\"\"\"F)!\"\"" } {TEXT -1 42 " is obtained by translating the graph of " }{XPPEDIT 18 0 "y = 5^x;" "6#/%\"yG)\"\"&%\"xG" }{TEXT -1 23 " downwards by one uni t." }}{PARA 0 "" 0 "" {TEXT -1 10 "The line " }{XPPEDIT 18 0 "y = -1; " "6#/%\"yG,$\"\"\"!\"\"" }{TEXT -1 28 " is a horizontal asymptote. " }}{PARA 0 "" 0 "" {TEXT -1 39 "In the following picture the graph of \+ " }{XPPEDIT 18 0 "y = log[5](x+1);" "6#/%\"yG-&%$logG6#\"\"&6#,&%\"xG \"\"\"F-F-" }{TEXT -1 15 " is plotted in " }{TEXT 368 3 "red" }{TEXT -1 22 ", while the graph of " }{XPPEDIT 18 0 "y = 5^x-1;" "6#/%\"yG,& )\"\"&%\"xG\"\"\"F)!\"\"" }{TEXT -1 15 " is plotted in " }{TEXT 260 4 "blue" }{TEXT -1 2 ". 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[\"F-7$Fajl$\"3%************>p\"F-7$7$Fajl$\"3@++++++KF-7$7$Fajl$\"3+++++++%)>F-F[cm7$F_cm7$Fajl$\"3s ***********f>#F-7$7$Fajl$\"3x***********fB#F-Fccm7$Fgcm7$Fajl$\"3]**** *******zW#F-7$7$Fajl$\"3a***********z[#F-F[dm7$F_dm7$Fajl$\"3G******** *****p#F-7$7$Fajl$\"3L************RFF-Fcdm7$Fgdm7$Fajl$\"31*********** >&HF-7$7$Fajl$\"35***********>*HF-F[em7$F_em7$Fajl$\"3&))**********R?$ F-7$7$Fajl$\"3*))**********RC$F-Fcem7$Fgem7$Fajl$\"3i)**********fX$F-7 $7$Fajl$\"3m)**********f\\$F-F[fm7$F_fm7$Fajl$\"3T)**********zq$F-7$7$ Fajl$\"3W)**********zu$F-Fcfm7$7$Fajl$\"3!*)**********zu$F-7$Fajl$\"3= )***********fRF-7$7$Fajl$\"3A)*************RF-F^gm-Fj[l6&F\\\\l$Fe[lFb jl$Fh[lFbjlFhgm-F\\jl6#\"\"$-F$6%7S7$FcjlFajl7$$!3)****\\(Gzni@F-Fajl7 $$!3')*\\7)GZ>V?F-Fajl7$$!3\")**\\7uL#)3>F-Fajl7$$!3o**\\()=8ct'***\\nD')4)*F*Fa jl7$$!3h'**\\PSPGg)F*Fajl7$$!3K'***\\_H,WsF*Fajl7$$!3k(***\\P!4'zeF*Fa jl7$$!3W'***\\ZIvkXF*Fajl7$$!3#z*\\7VytqLF*Fajl7$$!3;'***\\x#Q4&>F*Faj l7$$!3S[*****4==[(FjqFajl7$$\"3wW+v=*)44lFjqFajl7$$\"3u,++:JM*)=F*Fajl 7$$\"3q/](=k+\"[KF*Fajl7$$\"3\"e+Dc7f>a%F*Fajl7$$\"3#o+](=Z'>*eF*Fajl7 $$\"38.]Pz7pJrF*Fajl7$$\"3u++DcQ!*o%)F*Fajl7$$\"3%[+vo!>*y&)*F*Fajl7$$ \"3A+D\")35q16F-Fajl7$$\"3X+]([3*GP7F-Fajl7$$\"3))****\\X$*>s8F-Fajl7$ $\"3E+]P:F=/:F-Fajl7$$\"3L+v=7M)=j\"F-Fajl7$$\"3c+]ijKntF-Fajl7$$\"32++DO#4r.#F-Fajl7$$\"3!**\\(ozRPg@F-Fajl7$$ \"3)4+]F!\\8&H#F-Fajl7$$\"3,+v=HW$>U#F-Fajl7$$\"3m+v$*pkZaDF-Fajl7$$\" 3/++DT01%o#F-Fajl7$$\"3^+voaIs>GF-Fajl7$$\"3)3++!z@Q]HF-Fajl7$$\"3]+]7 0++%3$F-Fajl7$$\"3O+v=39^;KF-Fajl7$$\"3c++]DcFQLF-Fajl7$$\"3Y+]if6$yZ$ F-Fajl7$FezFajl7$$\"3g+v=*eNdt$F-Fajl7$$\"3=+D\"=C9J'QF-Fajl7$Fd[lFajl -Fj[l6&F\\\\lFhgmFhgmFggmFigm-%%TEXTG6%7$$\"$\\$F_\\l$!#:F_\\lQ\"x6\"- Fj[l6&F\\\\l$Fh[lF_\\lFjanFjan-F_an6%7$FdanFbanQ\"yFganFhan-F_an6%7$$F [hmFf[l$F_`lFbjlQ&y~=~xFganFhan-%%FONTG6$%*HELVETICAG\"\"(-%+AXESLABEL SG6%%!GF]cn-Ffbn6#%(DEFAULTG-%%VIEWG6$;$!$\\#F_\\lFbanFdcn" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 262 4 "Note" }{TEXT -1 83 ": The two gr aphs meet at the origin. The other point of intersection occurs where \+ " }{TEXT 462 1 "x" }{TEXT -1 1 " " }{TEXT 461 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "-0" "6#,$\"\"!!\"\"" }{TEXT -1 9 ".647016. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "Code for pictures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 19 "Code for pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f := x -> 2^x:\nmap(f,[10,2 0,50,100]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"%C5\"(w&[5\"1CE%o !***e7\"\"@w`?.n\\,%H#G-g]wE\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "f := x -> 2^x:\nmap(f,[-3, \+ -5/2,-2,-3/2, -1, -1/2, 0, 1/2, 1,3/2,2,5/2,3]);\nevalf(%):\nevalf[6]( %);\nmap(evalf@(x->[x,f(x)]),[seq(j/2,j=-6..6)]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "evalf(2^Pi); \n2^3.141593;\n2^3.14159;\n2^3141593:\nevalf(%);\nlog[10](%);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 663 "f := x -> 2^x:\np1 := plot(f(x),x=-3.4..3.4,thickness=2,color=COL OR(RGB,.93,0,0)):\np2 := plot([map(evalf@(x->[x,f(x)]),[seq(j/2,j=-6.. 6)])$4],style=point,symbol=[circle$2,diamond,cross],\n symbolsize=[ 15,10$3],color=[black,COLOR(RGB,0,1,0)$3]):\np3 := plot([seq([[i/2,-.3 ],[i/2,8.4]],i=-6..6),seq([[-3.4,j/2],[3.4,j/2]],j=0..16)],color=COLOR (RGB,.6,.6,.6)):\nt1 := plots[textplot]([[3.39,-.19,`x`],[-.2,8.39,`y` ]]):\nplots[display]([p||(1..3),t1],view=[-3.39..3.39,-.19..8.39],\n \+ scaling=unconstrained,font=[HELVETICA,7],\n xtickmarks=[-3=`-3`,-2=` -2`,-1=`-1`,0=0,1=`1`,2=`2`,3=`3`],\n ytickmarks=[1=`1`,2=`2`,3=`3`, 4=`4`,5=`5`,6=`6`,7=`7`,8=`8`],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 7 " circle" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1198 "x1 \+ := 1.5: y1 := evalf(2*sqrt(2)):\nf := x -> 2^x:\nx1 := 2.43: y1 := eva lf(f(x1)):\np1 := plot(f(x),x=-3.4..3.4,thickness=2,color=COLOR(RGB,.9 3,0,0)):\np2 := plot([[0,y1],[x1,y1],[x1,0]],color=black,linestyle=3): \np3 := plot([[[0,y1],[x1,y1],[x1,0]]$4],style=point,symbol=[circle$2, diamond,cross],\n symbolsize=[15,10$3],color=[black,COLOR(RGB,0,1,0 )$3]):\np4 := plottools[circle]([-.37,5.45],.25,color=COLOR(RGB,0,.4,0 )):\np5 := plottools[ellipse]([2.4,-.7],.9,.45, color=COLOR(RGB,0,.4,0 )):\np6 := plottools[arrow]([1.23,y1],[1.33,y1],0,.2,1,arrow):\np7 := \+ plottools[arrow]([x1,2.6],[x1,2.5],0,.15,1.3,arrow):\nt1 := plots[text plot]([[3.39,-.19,`x`],[-.2,8.39,`y`]],font=[HELVETICA,8]):\nt2 := plo ts[textplot]([[-.3,5.48,`y`]],color=COLOR(RGB,0,.4,0),font=[HELVETICA, 8]):\nt3 := plots[textplot]([[2.92,-.89,`2`]],color=COLOR(RGB,0,.4,0), font=[HELVETICA,7]):\nt4 := plots[textplot]([[2.4,-.7,`x = log y`]],c olor=COLOR(RGB,0,.4,0),font=[HELVETICA,8]):\nplots[display]([p||(1..7) ,t||(1..4)],view=[-3.39..3.39,-1.3..8.39],\n scaling=unconstrained,fo nt=[HELVETICA,8],\n xtickmarks=[-3=`-3`,-2=`-2`,-1=`-1`,0=0,1=`1`,2= `2`,3=`3`],\n ytickmarks=[1=`1`,2=`2`,3=`3`,4=`4`,5=`5`,6=`6`,7=`7`, 8=`8`],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "`plot/color`(navy);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1354 "f := \+ x -> 2^x:\np1 := plot([x,f(x),x=-4.4..3.4],thickness=2,color=COLOR(RGB ,.93,0,0)):\np2 := plot([f(x),x,x=-4.4..3.4],thickness=2,color=COLOR(R GB,.3,.1,.86)):\np3 := plot([[[0,1],[1,2]]$4],style=point,symbol=[circ le$2,diamond,cross],\n symbolsize=[15,10$3],color=[black,COLOR(RGB, 0,1,0)$3]):\np4 := plot(x,x=-4.4..8.4,color=COLOR(RGB,.3,.6,.3),linest yle=1,thickness=2):\np5 := plot([[[1,0],[2,1]]$4],style=point,symbol=[ circle$2,diamond,cross],\n symbolsize=[15,10$3],color=[black,COLOR( RGB,1,.5,0)$3]):\np6 := plot([seq([[i/2,-4.4],[i/2,8.4]],i=-8..16),seq ([[-4.4,j/2],[8.4,j/2]],j=-8..16)],\n color=COLOR(RGB,.6,.6,.6)):\nt1 \+ := plots[textplot]([[8.39,-.25,`x`],[.23,8.39,`y`],[6,5.2,`y = x`]]): \nt2 := plots[textplot]([3.5,7,`y = 2`],color=COLOR(RGB,.75,0,0),font= [HELVETICA,8]):\nt3 := plots[textplot]([3.9,7.2,`x`],color=COLOR(RGB,. 75,0,0),font=[HELVETICA,8]):\nt4 := plots[textplot]([7,2.3,`y = log \+ x`],color=COLOR(RGB,0,0,.7),font=[HELVETICA,8]):\nt5 := plots[textplot ]([7.35,2.1,`2`],color=COLOR(RGB,0,0,.7),font=[HELVETICA,8]):\nplots[d isplay]([p||(1..6),t||(1..5)],view=[-4.39..8.39,-4.39..8.39],\n scali ng=unconstrained,font=[HELVETICA,8],\n xtickmarks=[-4=`-4`,-3=`-3`,- 2=`-2`,-1=`-1`,1=`1`,2=`2`,3=`3`,4=`4`,5=`5`,6=`6`,7=`7`,8=`8`],\n y tickmarks=[-4=`-4`,-3=`-3`,-2=`-2`,-1=`-1`,1=`1`,2=`2`,3=`3`,4=`4`,5=` 5`,6=`6`,7=`7`,8=`8`],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 728 "f := x -> 2.5^x:\nx1 : = 1: y1 := evalf(f(x1)):\np1 := plot(f(x),x=-2.4..2.4,thickness=2,colo r=COLOR(RGB,.93,0,0)):\np2 := plot([[0,y1],[x1,y1],[x1,0]],color=black ,linestyle=3):\np3 := plot([[[0,1],[x1,y1]]$4],style=point,symbol=[cir cle$2,diamond,cross],\n symbolsize=[15,10$3],color=[black,COLO R(RGB,0,1,0)$3]):\nt1 := plots[textplot]([[2.39,-.19,`x`],[-.17,5.39,` y`],[-.2,1.05,1]],font=[HELVETICA,7]):\nt2 := plots[textplot]([[2.84,- .89,`2`]],color=COLOR(RGB,0,.4,0),font=[HELVETICA,7]):\nt3 := plots[te xtplot]([[-.2,y1,`a`]],font=[TIMES,ITALIC,8]):\nplots[display]([p||(1. .3),t||(1..3)],view=[-2.39..2.39,-.3..5.39],scaling=unconstrained,\n \+ font=[HELVETICA,8],xtickmarks=[1=`1`],ytickmarks=[y1=``], labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 766 "x1 := 1.5: y1 := evalf(2*sqrt(2)):\nf := x - > 2.5^(-x):\nx1 := 1: y1 := evalf(f(x1)):\np1 := plot(f(x),x=-1.7..2.4 ,thickness=2,color=COLOR(RGB,.93,0,0)):\np2 := plot([[0,y1],[x1,y1],[x 1,0]],color=black,linestyle=3):\np3 := plot([[[0,1],[x1,y1]]$4],style= point,symbol=[circle$2,diamond,cross],\n symbolsize=[15,10$3], color=[black,COLOR(RGB,0,1,0)$3]):\nt1 := plots[textplot]([[2.39,-.15, `x`],[-.13,3.89,`y`],[-.2,1.05,1]],font=[HELVETICA,7]):\nt2 := plots[t extplot]([[2.84,-.89,`2`]],color=COLOR(RGB,0,.4,0),font=[HELVETICA,7]) :\nt3 := plots[textplot]([[-.2,y1,`a`]],font=[TIMES,ITALIC,8]):\nplots [display]([p||(1..3),t||(1..3)],view=[-1.69..2.39,-.3..3.89],scaling=u nconstrained,\n font=[HELVETICA,8],xtickmarks=[1=`1`],yt ickmarks=[y1=``],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1358 "f := x -> 2.5^x:\nx1 := 1: y1 := evalf(f(x1)):\np1 := plot([x,f(x),x=-2.4..2.4],thickness=2,colo r=COLOR(RGB,.93,0,0)):\np2 := plot([f(x),x,x=-2.4..2.4],thickness=2,co lor=COLOR(RGB,.3,.1,.86)):\np3 := plot(x,x=-4.4..8.4,color=COLOR(RGB,. 3,.6,.3),linestyle=1,thickness=2):\np4 := plot([[[0,1],[x1,y1]]$4],sty le=point,symbol=[circle$2,diamond,cross],\n symbolsize=[15,10$ 3],color=[black,COLOR(RGB,0,1,0)$3]):\np5 := plot([[[1,0],[y1,x1]]$4], style=point,symbol=[circle$2,diamond,cross],\n symbolsize=[15, 10$3],color=[black,COLOR(RGB,1,.5,0)$3]):\nt1 := plots[textplot]([[5.3 9,-.17,`x`],[-.17,5.39,`y`],[-.2,1.05,1],[1.05,-.2,1],\n [ 4,3.5,`y = x`]],font=[HELVETICA,7]):\nt2 := plots[textplot]([[-.2,y1,` a`],[y1,-.2,`a`]],font=[TIMES,ITALIC,8]):\nt3 := plots[textplot]([2,4, `y =`],color=COLOR(RGB,.75,0,0),font=[HELVETICA,8]):\nt4 := plots[text plot]([2.37,4.02,`a`],color=COLOR(RGB,.75,0,0),font=[TIMES,ITALIC,8]): \nt5 := plots[textplot]([2.55,4.1,`x`],color=COLOR(RGB,.75,0,0),font=[ HELVETICA,8]):\nt6 := plots[textplot]([4,2,`y = log x`],color=COLOR( RGB,0,0,.7),font=[HELVETICA,8]):\nt7 := plots[textplot]([4.25,1.9,`a`] ,color=COLOR(RGB,0,0,.7),font=[TIMES,ITALIC,8]):\nplots[display]([p||( 1..5),t||(1..7)],view=[-2.39..5.39,-2.39..5.39],scaling=unconstrained, \n font=[HELVETICA,8],xtickmarks=[y1=``],ytickmarks=[y1= ``],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 805 "f := x -> log[10](x):\np1 := plot(f(x),x =0.01..11.89,thickness=2,color=COLOR(RGB,.93,0,0)):\np2 := plot([map(e valf@(x->[x,f(x)]),[seq(j,j=1..10)])$4],style=point,symbol=[circle$2,d iamond,cross],\n symbolsize=[15,10$3],color=[black,COLOR(RGB,0,1,0) $3]):\np3 := plot([seq([[i,-.25],[i,1.3]],i=0..12),seq([[-.1,j/10],[13 ,j/10]],j=-2..12)],color=COLOR(RGB,.6,.6,.6)):\nt1 := plots[textplot]( [[12.69,-.05,`x`],[-.25,1.29,`y`]]):\nplots[display]([p||(1..3),t1],vi ew=[-.25..12.69,-.25..1.29],\n scaling=unconstrained,font=[HELVETICA, 7],\n xtickmarks=[0=0,1=`1`,2=`2`,3=`3`,4=`4`,5=`5`,6=`6`,7=`7`,8=`8 `,9=`9`,10=`10`,11=`11`,12=`12`],\n ytickmarks=[-.2=`-0.2`,-.1=`-0.1 `,.1=`0.1`,.2=`0.2`,.3=`0.3`,.4=`0.4`,.5=`0.5`,.6=`0.6`,.7=`0.7`,.8=`0 .8`,.9=`0.9`,1=`1.0`,\n 1.1=`1.1`,1.2=`1.2`],labels=[``, ``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 843 "f := x -> log[2](x):\np1 := plot(f(x),x=0.01..11.89, thickness=2,color=COLOR(RGB,.93,0,0)):\np2 := plot([map(evalf@(x->[x,f (x)]),[seq(j,j=1..10)])$4],style=point,symbol=[circle$2,diamond,cross] ,\n symbolsize=[15,10$3],color=[black,COLOR(RGB,0,1,0)$3]):\np3 := \+ plot([seq([[i,-.25],[i,3.8]],i=0..12),seq([[-.1,j/5],[13,j/5]],j=-2..1 7)],color=COLOR(RGB,.6,.6,.6)):\nt1 := plots[textplot]([[12.69,-.08,`x `],[-.25,3.59,`y`]]):\nplots[display]([p||(1..3),t1],view=[-.25..12.69 ,-.25..3.59],\n scaling=unconstrained,font=[HELVETICA,7],\n xtickma rks=[0=0,1=`1`,2=`2`,3=`3`,4=`4`,5=`5`,6=`6`,7=`7`,8=`8`,9=`9`,10=`10` ,11=`11`,12=`12`],\n ytickmarks=[-.2=`-0.2`,0=`0`,.2=`0.2`,.4=`0.4`, .6=`0.6`,.8=`0.8`,1=`1.0`,1.2=`1.2`,1.4=`1.4`,1.6=`1.6`,1.8=`1.8`,2=`2 .0`,\n 2.2=`2.2`,2.4=`2.4`,2.6=`2.6`,2.8=`2.8`,3=`3.0`,3.2=`3 .2`,3.4=`3.4`],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 407 "f := x -> 3^(x-2):\np1 := p lot([x,f(x),x=-1.3..4],color=COLOR(RGB,.93,0,0),thickness=2):\np2 := p lot([f(x),x,x=-1.3..4],color=COLOR(RGB,.26,.12,.67),thickness=2):\np3 \+ := plot(x,x=-0..4.5,color=black,linestyle=2):\nt1 := plots[textplot]([ [4.49,-.1,`x`],[-.1,4.49,`y`],[4.4,3.9,`y = x`]],color=COLOR(RGB,.01,. 01,.01)):\nplots[display]([p||(1..3),t1],font=[HELVETICA,7],labels=[`` ,``],view=[-1.3..4.49,-1.3..4.49]);" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 570 "f := x -> log[5](x+1):\np 1 := plot([x,f(x),x=-2.3..4],color=COLOR(RGB,.93,0,0),thickness=2):\np 2 := plot([f(x),x,x=-2.3..4],color=COLOR(RGB,.26,.12,.67),thickness=2) :\np3 := plot(x,x=-0..4.5,color=black,linestyle=2):\np4 := plots[impli citplot](\{x=-1\},x=-2.3..4,y=-2.3..4,color=COLOR(RGB,.4,.1,.1),linest yle=3):\np5 := plot(-1,x=-2.3..4,color=COLOR(RGB,.1,.1,.4),linestyle=3 ):\nt1 := plots[textplot]([[3.49,-.15,`x`],[-.15,3.49,`y`],[3,2.6,`y = x`]],color=COLOR(RGB,.01,.01,.01)):\nplots[display]([p||(1..5),t1],fo nt=[HELVETICA,7],labels=[``,``],view=[-2.49..3.49,-2.49..3.49]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }