Understanding Logarithms: Exponential Growth, Properties, and Applications

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

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

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

26. 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

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

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

29. 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.

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 2log2(x)? • the result I get from raising 2 to the power I need to raise 2 to to get x = x.

31. 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

32. 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

33. 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

34. 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

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

36. Properties of logs

37. Exponential Property adding multiplying 23+4=2324

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

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

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

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

42. 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

43. 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)

44. 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)

45. 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)

46. 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)

47. 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)

48. 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

49. What about roots?

50. 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