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Properties of logs The basic property of logarithims • Loga(bc)=logab+Logac Example • • • • • Loga(b4) =loga(bbbb) =loga(b)+Loga(bbb) =loga(b)+Loga(b) +Loga(b) +Loga(b) =4Loga(b) The basic properties of logarithims • Loga(bc)=logab+Logac • Loga(bn)=n*logab Example • • • • • • • • x=log832 what is x? Rewrite as an exponential equation 8x=32 Take log2 of both sides Log2(8x)=Log232 xLog2(8)=Log232 x=Log2(32)/Log2(8) x=5/3 Change of base • • • • • • • • x=logay what is x? Rewrite as an exponential equation ax=y Take logc of both sides Logc(ax)=Logcy xLogc(a)=Logcy x=Logc(y)/Logc(a) logay=Logc(y)/Logc(a) Change of base • Using this rule on your calculator • logay=Logc(y)/Logc(a) If you’re looking for the logay use… Log(y)÷Log(a) Or ln(y)÷ln(a) The basic properties of logarithims • Loga(bc)=logab+Logac • Loga(bn)=n*logab • Logab=logc(b)/logc(a) Side effect: you only ever need one log button on your calculator. Logab=log(b)/log(a) Logab=ln(b)/ln(a) Warning: Remember order of operations WRONG log(2ax) =x*log(2a) =x*[log(2)+log(a)] =x log 2 + x log a CORRECT log(2ax) =log(2(ax)) =log(2)+log(ax) =log(2)+x*log(a) What about division? • • • • • • Loga(b/c) =Loga(b(1/c)) =Loga(bc-1) =Loga(b) + Loga(c-1) =Loga(b) + -1*Loga(c) =Loga(b) - Loga(c) The advanced properties of logarithims • Loga(bc)=logab+Logac • Loga(bn)=n*logab • Logab=logc(b)/logc(a) Side effect: you only ever need one log button on your calculator. Logab=log(b)/log(a) Logab=ln(b)/ln(a) Loga(b/c)=logab-Logac What about roots? log a ( n b ) 1/n log a (b ) 1 log a (b) n The advanced properties of logarithims • Loga(bc)=logab+Logac • Loga(bn)=n*logab • Logab=logc(b)/logc(a) Side effect: you only ever need one log button on your calculator. Logab=log(b)/log(a) Logab=ln(b)/ln(a) Loga(b/c)=logab-Logac Loga(n√b̅)=[logab]/n REVIEW QUESTION Simplify: log4(1/256) a) b) c) d) e) -2.40824 -5.5452 -.25 4 -4 REVIEW QUESTION Simplify: log4(1/256) There are lots of ways to do this. Here’s how I did it. log4(1/256) =log4(1)-log4(256) =0-log4(256) =0-log4(28) =0-8*log4(2) =0-8*log4(√4) =0-8*log4(4½) =0-8*(½) =-4 E Simplifying log expressions Expand • • • • Log(3x4/√y) =Log(3)+Log(x4)-Log(√y) =Log(3)+Log(x4)-Log(y½) =Log(3)+4*Log(x)-½Log(y) • Step 1: Expand * into + and ÷ into – • Step 2: convert nth roots into 1/n powers • Step 3: Expand ^ into * Condense • • • • Log(7)-2Log(x)+¾Log(q) Log(7)-Log(x2)+Log(q¾) Log(7/x2)+Log(q¾) Log(7q¾/x2) • Step 1: Condense * into ^ • Step 2: Condense – into ÷ • Step 3: Condense + into * Condense using the properties of logarithms: 3log(x) -2log(y) a) b) c) d) e) 3 2 log(x y ) 3 2 log(x ) log(y ) 3 2 log(x )/ log(y ) 3 2 log(x - y ) None of the above Condense using the properties of logarithms: 3log(x) -2log(y) 3log(x) -2log(y) Condense * into ^ log(x3) -log(y2) Condense – into ÷ log(x3/y2) E. None of the above. Solving Exponential Equations And log equations, too, I guess Example Solve: Add 7 to both sides 3x+2 - 7 = 12 3 x+2 = 19 Take the log of both sides log(3x+2 ) = log(19) Pull out the exponent (x+2)log(3) = log(19) Divide both sides by log(3) x+2 = log(19) / log(3) Subtract 2 from both sides x=log(19)/log(3)-2 Use calculator if you want x » 0.680144 General Strategy • An exponential equation has a variable in the exponent • Get the exponent part by itself • Take the log() of both sides – Or if you want, take the ln() of both sides • Use properties of logs to pull the power out. • Solve for your variable Example with ln() Solve: Divide both sides by 4 Take the ln() of both sides Pull the exponent out ln(e)=1 Solve for x 4e 2 x-7 = 12 e 2 x-7 = 3 ln(e 2 x-7 ) = ln(3) (2x - 7)ln(e) = ln(3) 2x - 7 = ln(3) x = [ln(3) + 7] / 2 Solving log equations Example Solve: log3 (2x + 7) + log3 (4) = 2 Combine logs log3 (4(2x + 7)) = 2 log3 (8x + 28) = 2 Distribute Exponentiate both sides 3log3 (8x+28) = 32 Cancel inverse functions 8x + 28 = 32 Solve for x 8x + 28 = 9 x = (9 - 28) / 8 x = -2.5 General strategy • • • • Combine logs to get one log by itself Exponentiate both sides with the matching base Exponential and log functions will cancel Solve for x Solve: a) b) c) d) e) x =5/e x = ln(e) x = ln(5) x=5 None of the above Solve: ln(ex)=ln(5) x*ln(e)=ln(5) x*1=ln(5) x=ln(5) C