1. Solving Exponential and Logarithm Equations
PreCalculus 3-4

2. Exponential & Log Equations

3. Exponential & Log Equations

4. Exponential & Log Equations
Find the solution to the equation, correct to three decimal places
x=-1.535
Exponential & Log Equations

5. Exponential & Log Equations

6. Exponential & Log Equations
Solve the following equation, correct to three decimal places
x=0.549
Exponential & Log Equations

7. Exponential & Log Equations

8. Exponential & Log Equations

9. Exponential & Log Equations
Solve the equation.
x=1.951
Exponential & Log Equations

10. Exponential & Log Equations

11. Exponential & Log Equations
Solve the equation.
x=0.693
Exponential & Log Equations

12. Exponential & Log Equations

13. Exponential & Log Equations
Solve the equation.
x=0
x=5
Exponential & Log Equations

14. Exponential & Log Equations

15. Exponential & Log Equations

16. Exponential & Log Equations

17. Exponential & Log Equations
Solve the equations
x=20.086
x=14
Exponential & Log Equations

18. Exponential & Log Equations

19. Exponential & Log Equations
Solve the equation x=
Exponential & Log Equations

20. Exponential & Log Equations

21. Exponential & Log Equations

22. Exponential & Log Equations
Solve the equation
x=2
Exponential & Log Equations

23. Exponential & Log Equations
Solve the equation
x=1001
Exponential & Log Equations

24. Exponential & Log Equations

25. Exponential & Log Equations
Solve the equation graphically
x=-3.101
x=5.997
Exponential & Log Equations

26. Exponential & Log Equations

27. Exponential & Log Equations

28. Exponential & Log Equations
Recall
Exponential & Log Equations

29. Exponential & Log Equations

30. Exponential & Log Equations

31. Exponential & Log Equations

32. Exponential & Log Equations

33. Exponential & Log Equations
Exploration
How long is it going to take to double my money?
Exponential & Log Equations

34. How long is it going to take to double my money?
Consider an investment of $100 invested at 5%, compounded continuously. How long would it take for the investor to have $200?

35. How long is it going to take to double my money?
What would the doubling time be if the initial investment were $1,000? $10,000? What effect does changing the principal have on the doubling time, and why?
The doubling time is not affected by changes in principal. Algebraically, the Po drops out of the exponential growth equation. Intuitively, the doubling time is a property of the ratio of two numbers, not a property of the numbers themselves.

36. How long is it going to take to double my money?
One of the first things that is taught in an economics class is the Rule of 72. It can be summarized:
"The number of years it takes an investment to double is equal to 72 divided by the annual percentage interest rate."

37. How long is it going to take to double my money?
What would the Rule of 72 say the doubling time of a 5% investment is? Is it a good estimate?
14.4 years
3.87% error

38. How long is it going to take to double my money?
Repeat for an investment of $100 at an interest rate of 3%, 8%, 12% and 18%. What can you say about the accuracy of the Rule of 72?
Interest Rate
Actual Doubling Time
Estimated Doubling Time
Error 3%
23.1 years
24 years
3.90% 8%
8.66 years
9 years
3.92%
12%
5.78 years
6 years
3.81%
18%
3.85 years
4 years

39. How long is it going to take to double my money?
Derive a precise formula for the time T to double an initial investment.

40. How long is it going to take to double my money?
There is an integer that gives a more accurate answer for continuous or nearly continuous compounding than the Rule of 72.
What is this number?
Check your answer by using it to estimate the doubling time of a 5% investment. 69
13.8 years
0.4% error

41. How long is it going to take to double my money?
It turns out that there is a reason that we use the number 72 in the Rule. It has to do with one of the assumptions we made.
Why do economists use the Rule of 72?
If the compounding is assumed to be annual, 72 works best for values of r between 3% and 15%. For example, 10% compounded annually doubles every years. The rule of 72 gives 7.2, a good approximation, and the rule of 69 gives 6.9, a bad one.

42. Exponential & Log Equations
Homework pg
18, 24, 28, 32, 34, 38, 48, 50, 52, 58, 76, 80, 82, 84, 103, 106, 109, 110, 112, 115, 121
Exponential & Log Equations