{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 49 "High School Modul es > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 259 10 "Logarithms" }}{PARA 0 "" 0 "" {TEXT -1 201 "\nExplorati on of logs - their concept, relationship to exponential equations, com mon and natural logs, computing other logs, simplify log expressions, \+ solving log equations, and graphing log functions.\n" }}{PARA 0 "" 0 " " {TEXT 260 153 "[Directions : Execute the Code Resource section first . Although there will be no output immediately, these definitions are \+ used later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 " 0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(plots): " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 89 "txln := `\\n_______________________________________ _____________________________________`:" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 " 1. The Idea of Logs" }}{PARA 0 "" 0 "" {TEXT -1 73 "\nWh en dealing with numbers which are different powers of the same base : \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a^k $ k = 1..12;" }}} {PARA 0 "" 0 "" {TEXT -1 111 "\nIt makes more sense to deal only with \+ the exponents at times. These are the logs of the original expressions .\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "k $ k = 1..12: `expo nents `= %;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'' log[a]( a^k)' = k ' $ k = 1..12;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'' log[a](a^k)' = k ' $ k = 1..12;" }}}{PARA 0 "" 0 "" {TEXT -1 42 "\nNegative powers means negative exponents." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 98 "a^(-k) $ k = 1..10: `Powers of 7 ` = %;\n( -k) $ k = 1..10: `Logs of these numbers (base 7) ` = %;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "7^k $ k = 1..10: `Powers of 7 ` \+ = %;\nsimplify(log[7](7^k)) $ k = 1..10: `Logs of these numbers (base \+ 7) ` = %;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "7^(-k) $ k = \+ 1..10: `Negative Powers of 7 ` = %;\nsimplify(log[7](7^(-k))) $ k = \+ 1..10: `Logs of these numbers (base 7) ` = %;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 156 "\nThe log is the ex ponent - when you take a base number and raise it to an exponent. Ther efore each log is depends on two numbers - the base and the result.\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "exponent := 7;\nbase := 19;\n`base^exponent` = base^exponent;\n`log of result `= round(log[19 ](base^exponent));\n`It's the same!`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "exponent := 4.5;\nbase := 3;\n`base^exponent` = base ^exponent;\n`log of result `= log[3](base^exponent);\n`It's the same!` ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "exponent := 3.52;\n6^e xponent;\n`log of result `= log[6](%);\n`It's the same!`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "exponent := 11.0892;\n134^exponent; \n`log of result `= log[134](%);\n`It's the same!`;" }}}{PARA 0 "" 0 " " {TEXT -1 166 "\nAs we noticed above, the log is depends on two numbe rs - the base and the result. The name resulting number has two differ ent logs, relative to two different bases.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "Result := 64:\nfor k from 1 to 12 do\nbase:=2^k: \n print(cat(`The log of 64 (base `, base, `) is `,simplify(log[base ](64))));\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 " " 0 "" {TEXT -1 33 " 2. Log & Exponential Conversion" }}{PARA 0 "" 0 "" {TEXT -1 246 "\nOne of the most fundamental transformation is betwe en log equations and exponential equations. At times we need to conver t one to the other, or vice-versa.\n\nThese two equations are equivila nt with the same letters representing the same values :\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "a^b = c;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "log[a](c) = b;" }}}{PARA 0 "" 0 "" {TEXT -1 87 "The se two equations are equivilant with the same letters representing the same values :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "\"2^3\" = \+ 8; 'log[2](8)' = 3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "`N ote both equations have the same solution!`;\n'log[3](7)' = c; solve(% ,c): % = evalf(%,20);\n3^c = 7; solve(%,c): % = evalf(%,20);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "`Note both equations have \+ the same solution!`;\n'log[12](y)' = 3; solve(%,y);\n12^3 = y; s olve(%,y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 42 " 3. Two Special Logs : Common and Natural" }}{PARA 0 " " 0 "" {TEXT -1 123 "\nThere are two special cases of the logarithm th at used commonly :\n - when a logarithm has base 10, it is cal led a " }{TEXT 263 16 "common logarithm" }{TEXT -1 40 ". \n \+ Often we use the notation " }{TEXT 261 5 "log a" }{TEXT -1 106 " (with out an apparent base) to signify common log.\n\n - when a logar ithm has base e, it is called a " }{TEXT 264 17 "natural logarithm" } {TEXT -1 33 ".\n We use the notation " }{TEXT 262 4 "ln a" } {TEXT -1 135 " to denote natural logs.\n\nMaple has special log functi ons for each of these. Here is the common log - commonly used in scien ce classes." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "`Common log`; \n'log[10](10^13)': '%' = simplify(%); \n'log10(10^13)': '%' = sim plify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "`Common log`; \n'log[10](231)': '%' = evalf(simplify(%)); \n'log10(231)': '%' = \+ evalf(simplify(%)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "`C ommon log`; \n'log[10](.032)': '%' = evalf(simplify(%)); \n'log10(.03 2)': '%' = evalf(simplify(%));" }}}{PARA 0 "" 0 "" {TEXT -1 56 "\nHe re is the natural log, naturally used in mathematics." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "`Natural log`; \n'log[exp(1)](31)': '%' = evalf(simplify(%)); \n'ln(31)': '%' = evalf(simplify(%)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "`Natural log`; \n'log[ exp(1)](10)': '%' = evalf(simplify(%)); \n'ln(10)': '%' = e valf(simplify(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "`Nat ural log`; \n'log[exp(1)](.01)': '%' = evalf(simplify(%)); \n'ln(.01) ': '%' = evalf(simplify(%));" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 " 4. Rules of Logs" }} {PARA 0 "" 0 "" {TEXT -1 88 "\nThere are a number or rules for logs. L ets look at some of them and see some examples.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "print(cat(txln,`\\n1. Products`));\n'log[b](p *q) = log[b](p) + log[b](q)';print(txln);" }}}{PARA 0 "" 0 "" {TEXT -1 46 "\nNotice that we get the same answers for both!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "''log[7](13*41)'': % = e valf(eval(%));\n''log[7](13) +log[7](41)'': % = evalf(eval(%));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "print(cat(txln,`\\n2. Quoti ents`)); \n 'log[b](p/q)' = 'log[b](p) - log[b](q)'; ;print(t xln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "''log[7](13/41)'': % = evalf(eval(%));\n''log[7](13) - log[7](41)'': % = eva lf(eval(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "print(cat(t xln,`\\n3. Zero Power is One`));'log[b](1)'=0;print(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "''log[5](1)'': % = simplify(eval(% ));\n''log[22](1)'': % = simplify(eval(%)); \n''log[53](1)'': % = si mplify(eval(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "print(c at(txln,`\\n4. Power of One is Identity`));'log[b](b)' = 1;print(txln) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "''log[5](5)'': % = si mplify(eval(%));\n''log[22](22)'': % = simplify(eval(%)); \n''log[53 ](53)'': % = simplify(eval(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "print(cat(txln,`\\n5. Powers - same base`));'log[b](b^r) = r'; print(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "''log[4.71] (4.71^3.22)'': % = evalf(eval(%));\n3.22;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "print(cat(txln,`\\n6. Powers - different \+ bases`));\n 'log[b](p^r) = r*log[b](p)';print(txln);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "''log[2](4.71^3.22)'': \+ % = evalf(eval(%));\n''3.22 * log[2](4.71)'': % = ev alf(eval(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "print(cat( txln,`\\n7. Log Applied to Exponent`));\n 'b^log[b](a) = a';pr int(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "''9^log[9](7) '': % = evalf(eval(%));\n 7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "print(cat(txln,`\\nNote.......Sums : THIS IS NOT A R ULE - It's a common MISTAKE!`)); \n 'log[b](p + q) = log[b]( p + q)'; ;print(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 " ''log[7](13 + 41)'': % = evalf(eval(%));\n''log[7](13) + lo g[7](41)'': % = evalf(eval(%));\n`NOT the same!`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "''log[7](92 + 37)'': % = evalf( eval(%));\n''log[7](92) + log[7](37)'': % = evalf(eval(%));\n`NOT the same!`;" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 " 5. Computing Other Log Values" }}{PARA 0 "" 0 "" {TEXT -1 29 "\nNotice this \"coincidenc e\" :\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "'log[2](3)' = eva lf( log[2](3), 20); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ln( 3)/ln(2): % = evalf(%, 20);" }}}{PARA 0 "" 0 "" {TEXT -1 20 "\nHere is an example." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "exponent := \+ evalf(ln(7)/ln(5),20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "e valf( log[5](7), 20 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "r ound( 5^exponent );" }}}{PARA 0 "" 0 "" {TEXT -1 52 "\n\nNote that all three of these have the same value ;" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "log[a](b); log10(b)/log10(a);ln(b)/ln(a);" }}}{PARA 0 "" 0 "" {TEXT -1 106 "\n \nThis is a method that you can use to comp ute logs on a calculator - other than common and natural logs." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "'log[3](10)'; % = evalf(%); " }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 52 " 6. Simplifying Log Expressions & Verifying Answers" }} {PARA 0 "" 0 "" {TEXT -1 52 "\nUsing those rules we can simplify log e xpressions.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "log( x^2 * \+ y^10 /z^3); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "log( x^2 ) \+ + log( y^10) - log( z^3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "2*log(x) + 10*log(y) -3*log(z);" }}}{PARA 0 "" 0 "" {TEXT -1 492 " \n\nWe did all the work for ourselves. However, we can use Maple to ch eck if our answer is wrong. If we randonly select some numbers, we sho uld get the same results from the original expression and our final si mplified expression. This doesn't actually prove the answer is correct , but it gives us a strong indication. If the two expressions agree wi th various substitutions, we can be reasonably sure It's correct. If t here is a disagreement, we will know for sure that our solution is wro ng!\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "`\\nCheck .... \\n `;\nevalf( subs( \{x = 2, y = 3, z = 7\}, log( x^2 * y^10 /z^3) ));\n evalf( subs( \{x = 2, y = 3, z = 7\}, 2*log(x) + 10*log(y) -3*log(z))) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "`\\nCheck .... \\n`; \nevalf( subs( \{x = 10, y = 22, z = 47\}, log( x^2 * y^10 /z^3) )); \nevalf( subs( \{x = 10, y = 22, z = 47\}, 2*log(x) + 10*log(y) -3*log (z)));" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}{PARA 0 "" 0 "" {TEXT -1 26 "\n\nHere is another example\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "log( sqrt( (a+b)/c^5 ));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "log( ( (a+b)/c^5 )^(1/2) );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "(1/2)*log( (a+b)/c^5 );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 30 "(1/2)*( log(a+b) - 5*log(c) );" }}}{PARA 0 "" 0 "" {TEXT -1 29 "\nLets check this answer too.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "`\\nCheck .... \\n`;\nevalf( subs( \{a = 2, \+ b = 3, c = 7\}, log( sqrt( (a+b)/c^5 )) ));\nevalf( subs( \{a = 2, b \+ = 3, c = 7\}, (1/2)*( log(a+b) - 5*log(c) )));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 154 "`\\nCheck .... \\n`;\nevalf( subs( \{a = 31, \+ b = 4, c = 9\}, log( sqrt( (a+b)/c^5 )) ));\nevalf( subs( \{a = 31, b = 4, c = 9\}, (1/2)*( log(a+b) - 5*log(c) )));" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 " 7 Graphing \+ Logs" }}{PARA 0 "" 0 "" {TEXT -1 32 "\nHere are some basic log graphs \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot( ln(x), x = 0..10 0);" }}}{PARA 0 "" 0 "" {TEXT -1 147 "\nHere are the graphs of log[2] \+ ...log[10] on the same graph. Note they all pass through the same poin t (1,0) and have the same vertical asymptote.\n" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "plot( \{ seq(log[k](x) , k = 2..10)\} , x = 0. .10);" }}}{PARA 0 "" 0 "" {TEXT -1 32 "\nHere log graphs shifted right .\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot( \{ seq(ln(x - k ) , k = 0..10)\} , x = 0..30, y = -4..3);" }}}{PARA 0 "" 0 "" {TEXT -1 33 "\nHere are log graphs shifted up.\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "plot( \{ seq(ln(x) + k , k = -6..6)\} , x = 0..20) ;" }}}{PARA 0 "" 0 "" {TEXT -1 47 "\nHere is a log graph shifted 4 rig ht, and 3 up\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot( ln(x - 4) + 3 , x = 0..20);" }}}{PARA 0 "" 0 "" {TEXT -1 87 "\nWe can be tter see this shift by also plottting where the axes would get shifted also.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "plots[display]( plot( ln(x - 4) + 3 , x = 0..20), \n plot( [[4,-1],[ 4,6]], color = blue, linestyle = 2), \n plot( [[0,3],[2 0,3]], color = blue, linestyle = 2));" }}}{PARA 0 "" 0 "" {TEXT -1 2 " \n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 26 " 8. Solving Log Equation s" }}{PARA 0 "" 0 "" {TEXT -1 6 "\n\n " }{TEXT 258 11 " Example 1 \+ " }{TEXT -1 6 ": " }{XPPEDIT 18 0 "log[7](3*x+2) = 3;" "6#/-&%$log G6#\"\"(6#,&*&\"\"$\"\"\"%\"xGF-F-\"\"#F-F," }{TEXT -1 2 ".\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "log[7]( 2*x + 1) = 3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve( %, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "7^3 = 2*x + 1; solve( %,x);" }}} {PARA 0 "" 0 "" {TEXT -1 6 "\n " }{TEXT 257 10 "Example 2 " } {TEXT -1 6 ": " }{XPPEDIT 18 0 "log[4]( x + 2) + log[4](x - 2) = 2 " "6#/,&-&%$logG6#\"\"%6#,&%\"xG\"\"\"\"\"#F-F--&F'6#F)6#,&F,F-F.!\" \"F-F." }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " log[4]( x + 2) + log[4](x - 2) = 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(%, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "log[4]( (x+2)*(x-2) ) = 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "(x+2)*(x-2) = 4^2;\nexpand(%);\nlhs(%)-rhs(%);\nsolv e(%);" }}}}{PARA 0 "" 0 "" {TEXT 265 36 "\n \251 2002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 1" 26 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }