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Properties of logs

Properties of logs. The basic property of logarithims. Log a ( bc )= log a b+Log a c. Example. Log a (b 4 ) = log a ( bbbb ) = log a ( b)+ Log a ( bbb ) = log a (b)+ Log a ( b) + Log a (b ) + Log a (b ) =4Log a (b). The basic properties of logarithims. Log a ( bc )= log a b+Log a c

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Properties of logs

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  1. Properties of logs

  2. The basic property of logarithims • Loga(bc)=logab+Logac

  3. Example • Loga(b4) • =loga(bbbb) • =loga(b)+Loga(bbb) • =loga(b)+Loga(b) +Loga(b) +Loga(b) • =4Loga(b)

  4. The basic properties of logarithims • Loga(bc)=logab+Logac • Loga(bn)=n*logab

  5. 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

  6. 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)

  7. Change of base • Using this rule on your calculator • logay=Logc(y)/Logc(a) If you’re looking for the logayuse… Log(y)÷Log(a) Or ln(y)÷ln(a)

  8. 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)

  9. 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)

  10. 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)

  11. 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

  12. What about roots?

  13. 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

  14. REVIEW QUESTIONSimplify: log4(1/256) • -2.40824 • -5.5452 • -.25 • 4 • -4

  15. REVIEW QUESTIONSimplify: 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

  16. Simplifying log expressions

  17. 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 *

  18. 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 *

  19. Condense using the properties of logarithms: 3log(x) -2log(y) • log(x3y2) • log(x3) log(y2) • log(x3)/ log(y2) • log(x3 - y2) • None of the above

  20. 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.

  21. Solving Exponential Equations And log equations, too, I guess

  22. Example

  23. 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

  24. Example with ln()

  25. Solving log equations

  26. Example

  27. 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

  28. Solve: • x =5/e • x = ln(e) • x = ln(5) • x = 5 • None of the above

  29. Solve: ln(ex)=ln(5) x*ln(e)=ln(5) x*1=ln(5) x=ln(5) C

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