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Understanding Logarithms: Exponential Growth, Properties, and Applications

warren-wilder avatar
warren-wilder
52 diapositives · 782 Views · Nov 07, 2014

This comprehensive guide delves into the fundamentals of logarithms and their application in exponential growth scenarios, such as doubling money over time. Through various examples, readers will learn how to manipulate logarithmic and exponential equations, utilize logarithmic properties, and convert expressions between logarithmic and exponential forms. Key concepts include identifying the domain of logarithmic functions and rewriting equations for greater clarity. This resource aims to build a solid foundation in logarithms for practical financial and mathematical applications.

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Understanding Logarithms: Exponential Growth, Properties, and Applications
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  1. Logarithms The “I’m going to lie to you a bit” version

  2. Example • Every year I double my money $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  3. Example • If I know what time it is and want to know how much money I have $1 a=1(2t) a=# of $ t=# of yrs $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  4. Example • What if I know money and want to know time? $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  5. Example • How long would it take me to get $1,000,000? $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  6. Two sequences +1 +1 +1 +1 +1 +1 +1 +1 *2 *2 *2 *2 *2 *2 *2 *2

  7. Exponential *2 a=2t

  8. Logarithmic *2 t=log2a

  9. Exponential *3 a=3t

  10. Logarithmic *3 t=log3a

  11. Exponential The special base 10 *10 a=10t

  12. Logarithmic The special base10 *10 t=log10a t=log(a)

  13. Exponential The special base e *e a=et

  14. Logarithmic The special base e *e t=logea t=ln(a)

  15. Logarithmic The special base e e≈2.7182818284 This number makes calculus easier *e t=logea t=ln(a)

  16. Meanings • a=2t • I know t. • a is the result I get from raising 2 to the t power. • t=log2(a) • I know a. • t is the power I need to raise 2 to to get a.

  17. Example • a=23 • a is the result I get from raising 2 to the power 3. • That result is 8. a=8. • t=log2(8) • t is the power I need to raise 2 to to get 8. • Since 23=8, t=3.

  18. Exponential a=2t *2 No matter what power I use, the result is always positive

  19. Exponential a=2t *2 There is no power that can get me a negative number

  20. Logarithmic t=log2a *2 There is no power that can get me a negative number

  21. Logarithmic t=log2a *2 I can only find the powers of positive numbers

  22. Logarithmic t=log2a *2 I can only take the log of positive numbers

  23. Logarithmic t=log2a *2 The domain of this function called “log2” is a>0

  24. Logarithmic t=log2a *2 The domain of any log(whatever) is always whatever>0

  25. Example problem • Find the domain of 2log7(4x-3)+7x-9 Whatever is inside the log has to be >0. I can find an answer whenever 4x-3>0 x>3/4

  26. Find the domain: • x > 5/4 • x < 5/4 • x > -5/4 • x < -5/4 • None of the above

  27. Find the domain: Whatever is inside the log has to be >0 5-4x>0 5>4x 5/4>x b) x < 5/4

  28. Rewriting equations • y=bx x=logby • 2=3p p=log32 • q+3=79 9=log7(q+3) • 9=32x+1 2x+1=log39 • 7=e4 4=ln(7) or 4=loge(7) • x+2=102x-1 2x-1=log(x+2) or 2x-1=log10(x+2) • The result of the log is the exponent. • The result of the exponent is what goes inside the log.

  29. Meanings • a=2t • I know t. • a is the result I get from raising 2 to the t power. • t=log2(a) • I know a. • t is the power I need to raise 2 to to get a. • What is 2log2(x)? • the result I get from raising 2 to the power I need to raise 2 to to get x = x.

  30. Meanings • a=2t • I know t. • a is the result I get from raising 2 to the t power. • t=log2(a) • I know a. • t is the power I need to raise 2 to to get a. • What is log2(2x)? • The power that I need to raise 2 to so that I get the result of raising 2 to the x power. =x

  31. Rewriting equations version 2 • 9=32x+1 • Taking the log of both sides. • Log3(9)=Log3(32x+1) • Log39=2x+1 • Exponentiating both sides • 3Log3(9)=32x+1 • 9=32x+1

  32. Rewriting equations version 2 • 3x-7=e2x+1 • Taking the log of both sides. • ln(3x-1)=ln(e2x+1) • ln(3x-7)=2x+1 • Exponentiating both sides • eln(3x-7)=e2x+1 • 3x-7=e2x+1

  33. Convert the following logarithmic expression into exponential form: y = ln(x+2). a) ey= x+2 b) 10y = x+2 c) e(x+2) = y d) 10 (x+2) = y e) None of the above

  34. Convert the following logarithmic expression into exponential form: y = ln(x+2). y = ln(x+2) ey=eln(x+2) ey=x+2 A

  35. Properties of logs

  36. Exponential Property adding multiplying 23+4=2324

  37. Logarithmic Property adding multiplying Log(8*16)=log(8)+log(16)

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

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

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

  41. 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

  42. 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)

  43. 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)

  44. 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)

  45. 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)

  46. 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)

  47. 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

  48. What about roots?

  49. 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