1. 1. Exponential GrowthExponential Growth 2. Exponential DecayExponential Decay 3. ee 4. Logarithmic FunctionsLogarithmic Functions 5. Properties of LogarithmsProperties of Logarithms 6. Solving Exponential and LogarithmicSolving Exponential and Logarithmic Equations A. appendixappendix Exponential Growth, Decay, e, and Logs

2. Exponential Growth and Decay Menu You invest $10,000 in a mutual fund that pays 6% interest each year. How much will your investment be worth in 1 year? What if you leave that investment in for 7 years?

3. Exponential Growth and Decay Menu You invest $10,000 in a mutual fund that pays 6% interest each year. How much will your investment be worth in 7 years? A = Amount (new value) P = Principle (original value) r = Rate (appreciation or depreciation as a decimal) T = Time (How many times interest is calculated) 1+R is also sometimes called the Growth Factor or Decay Factor

4. Exponential Growth and Decay Menu COMPOUND INTEREST: You invest $10,000 in a mutual fund that pays 6% interest per year, compounded monthly. How much will that investment be worth in 7 years? This is almost the same formula. “n” represents how many times the interest is recalculated in a cycle.

5. Exponential Growth and Decay Menu COMPOUND INTEREST: You invest $10,000 in a mutual fund that pays 6% interest per year, compounded monthly. How much will that investment be worth in 7 years? STANDARD (Simple) INTEREST: You invest $10,000 in a mutual fund that pays 6% interest each year. How much will your investment be worth in 7 years?

6. Exponential Growth and Decay Menu STANDARD INTEREST (decay / depreciation): You buy a $50,000 sports car. That car depreciates at a rate of 15% per year. How much will the car be worth after 8 years?

7. Exponential Growth and Decay Menu STANDARD INTEREST (decay / depreciation): You buy a $50,000 sports car. That car depreciates at a rate of 15% per year. How much will the car be worth after 8 years? Graph the value of the car over time:

8. Exponential Growth and Decay Menu Notice: as we go farther to the right, the graph gets closer to 0 but never reaches it. This is an asymptote. Y=0

9. Exponential Growth and Decay Menu GRAPHING growth and decay functions Principle Base or Growth Factor Asymptote Exponent

10. Exponential Growth and Decay Menu GRAPHING growth and decay functions XYXY 0 1 Always pick the number that will make the exponent zero 3 5 9 Y = 1

11. Exponential Growth and Decay Menu GRAPHING growth and decay functions XYXY 5 4 6 -2.5 2 Y = -4

12. Exponential Growth and Decay Menu Y = -4 Y = 1 Domain: Range: Domain: Range: All Real Y > 1 All Real Y > -4

13. “e” N= 10 100 1,000 10,000 Hey, how’d you put infinity in? CALCULUS e Menu

14. So what’s the deal with “e” e Menu

15. So what’s the deal with “e” The growth problems you have done were all “annual appreciation”, “compounded daily” or “decreases 5% per year” annual daily year e Menu

16. So what’s the deal with “e” But things that constantly grow or “happen all the time” grow at the natural exponential rate: What is to circles, e is to growth. It just how things grow. e Menu

17. So what’s the deal with “e” What grows/shrinks continuously? Life: populations Bacteria Cells Rabbits People Virus Shrubberies Radioactive stuff e Menu

18. You invest $100 at 6% annual interest. After 3 years how much will you have if it is compounded once per year? P= r = t = 100 0.06 3 e Menu

19. You invest $100 at 6% annual interest. After 3 years how much will you have if it is compounded quarterly (4 times) per year? P= r = t = n = 100 0.06 3 4 e Menu

20. You invest $100 at 6% annual interest. After 3 years how much will you have if it is compounded monthly (12 times) per year? P= r = t = n = 100 0.06 3 12 e Menu

21. You invest $100 at 6% annual interest. After 3 years how much will you have if it is compounded CONTINUOUSLY? P= r = t = 100 0.06 3 e Menu

22. Compounded Monthly Compounded quarterly Standard interest Compounded continuously e Menu

23. How to use “e” If something is growing continuously: e Menu

24. Compounded continuously: Compounded “n” times per time span Regular interest Given the half-life Rule of 72 time it takes to double or half e Menu

25. e Some Practice Problems:

26. e Menu Some Practice Problems: If the base is more than 1, it is growth. If the base is less than 1, it is decay. growth decay decay

27. e Menu Some Practice Problems:

28. e Menu Some Practice Problems:

29. Logarithmic Functions Logarithms are a way of asking for an exponent. Use this formula to rewrite each of the following as a logarithmic equation Menu

30. Logarithmic Functions Use this formula to rewrite each of the following as a Exponential equation Menu

31. Logarithmic Functions Menu Some common shortcuts:

32. Logarithmic Functions Menu Some Practice Problems: What power do we raise 5 to, in order to get 25?

33. Logarithmic Functions Menu Some Practice Problems:

34. Logarithmic Functions Menu Some Practice Problems:

35. Logarithmic Functions Menu Some Practice Problems:

36. Logarithmic Functions Menu Some Practice Problems:

37. Logarithmic Functions Menu Finding the inverse of a logarithm Finding the inverse of a Logarithmic function is the same for any other function… Example Find the inverse of this function Just switch x and y, then resolve for y OR

38. Logarithmic Functions Menu Problems Find the inverse 1.Switch X and Y 2.Rewrite as an exponential equation 3. Solve for Y

39. Properties of Logarithms Menu There are 3 basic rules for logarithms that we will use to expand or condense a logarithmic expression: Problems Expand this expression Product Property Quotient Property Exponent Property This is the type of problem we want to be able to do.

40. Properties of Logarithms Menu Product Property Examples Expand the expressions

41. Properties of Logarithms Menu Examples Expand the expressions Quotient Property

42. Properties of Logarithms Menu Examples Expand the expressions Exponent Property

43. Properties of Logarithms Menu Problems Expand these expressions

44. Properties of Logarithms Menu Problems Condense these expressions

45. Properties of Logarithms Menu The change of base formula In order to evaluate a logarithm with a base that is not 10 or e… (In other words, you can’t just type it in your calculator, but you want a numeric answer) You can evaluate it using the CHANGE OF BASE formula Table estimate?

46. Properties of Logarithms Menu The change of base formula

47. Solving Exponential and Logarithmic Functions Menu Solve for X. Since the two sides are equal, we can take the log of both. Example 1A

48. Solving Exponential and Logarithmic Functions Menu Solve for X. Since 8 and 4 are both powers of 2… We can solve this without the calculator Rewrite both as a power of 2 Simplify If they are equal, the exponents must be equal! Example 1B

49. Solving Exponential and Logarithmic Functions Menu Solve for X. Example 2 OR

50. Solving Exponential and Logarithmic Functions Menu Solve for X. Example 3

51. Solving Exponential and Logarithmic Functions Menu Solve for X. Example 4

52. Solving Exponential and Logarithmic Functions Menu Solve for X. Example 5 X

53. Solving Exponential and Logarithmic Functions Menu Solve for X. 1.2. X X