1. GRAPHING EXPONENTIAL FUNCTIONS f(x) = 2 x 2 > 1 exponential growth 2 24–2 4 6 –4 y x Notice the asymptote: y = 0 Domain: All real, Range: y > 0

2. GRAPHING EXPONENTIAL DECAY 0 < < 1 exponential decay 1 2 2 24–2 4 6 –4 y x Notice the asymptote: y = 0 Domain: All real, Range: y > 0

3. Graph Natural Base Exponential Functions Use the graph of f ( x ) = e x to describe the transformation that results in h ( x ) = e –x – 1. Then sketch the graph of the function. Answer: h (x) is the graph of f (x) reflected in the y-axis and translated 1 unit down with a vertical asymptote at x = -1. Domain: All real, Range: y > -1

4. Graph f(x) = e x–2 + 1. Graphing Exponential Functions VA: x = 1 Domain: All real, Range: y > 1

5. COMPOUND INTEREST FORMULA A : amount of the investment at time t P : principal r : annual interest rate as a decimal n : number of times interest is compounded per year t : time in years A(t) = P 1 + () r n nt

6. FIND THE FINAL AMOUNT OF $100 INVESTED AFTER 10 YEARS AT 5% INTEREST COMPOUNDED ANNUALLY, QUARTERLY AND DAILY. ANS: $162.89 annually, $164.36 quarterly, $164.87 daily

7. Recall the compound interest formula A = P(1 + ) nt, where A is the amount, P is the principal, r is the annual interest, n is the number of times the interest is compounded per year and t is the time in years. n r The formula for continuously compounded interest is A = Pe rt, where A is the total amount, P is the principal, r is the annual interest rate, and t is the time in years.

8. What is the total amount for an investment of $500 invested at 5.25% for 40 years and compounded continuously? Economics Application The total amount is $4083.08. A = Pe rt Substitute 500 for P, 0.0525 for r, and 40 for t. A = 500e 0.0525(40) Use the e x key on a calculator. A ≈ 4083.08

9. You can write an exponential equation as a logarithmic equation and vice versa.

10. Logarithmic Form Exponential Equation log 9 9 = 1 log 2 512 = 9 log 8 2 = log 4 = –2 log b 1 = 0 1 16 1 3 9 1 = 9 2 9 = 512 1 3 8 = 2 1 16 4 –2 = b 0 = 1

11. Evaluate by using mental math. Evaluating Logarithms by Using Mental Math The log is the exponent. Think: What power of 5 is 125 ? log 5 125 5 ? = 125 5 3 = 125 log 5 125 = 3

12. Exponential and logarithmic operations undo each other since they are inverse operations.

13. Simplify. a. ln e 3.2 b. e 2lnx c. ln e x +4y ln e 3.2 = 3.2 e 2lnx = x 2 ln e x + 4y = x + 4y

15. Graphs of Logarithmic Functions Sketch and analyze the graph of f ( x ) = log 2 x. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

16. Graphs of Logarithmic Functions Answer: Domain: (0, ∞); Range: (–∞, ∞); x-intercept: 1; Asymptote: y-axis; Increasing: (0, ∞); End behavior: ;

18. Expand Logarithmic Expressions A. Expand ln 4 m 3 n 5. Answer: ln 4 + 3 ln m + 5 ln n

19. Expand Logarithmic Expressions Expand.

20. Condense Logarithmic Expressions Condense. Answer:

22. Use the Change of Base Formula Evaluate log 6 4. log 6 4 =Change of Base Formula ≈ 0.77Use a calculator. Answer: 0.77

23. Solve Logarithmic Equations Using One-to-One Property Solve 2 ln x = 18. Give exact and round to the nearest hundredth. 2 ln x= 18 ln x= 9 e ln x = e 9 x= e 9 x≈ 8103.08

24. Solve 7 – 3 log 10 x = 13. Round to the nearest hundredth. 7 – 3 log 10x= 13 –3 log 10x= 6 log 10x= –2 10 –2 =10x 10 –3 = x = x Log Circle at this point.

25. Solve log 2 5 = log 2 10 – log 2 ( x – 4). log 2 5= log 2 10 – log 2 (x – 4) log 2 5= 5=5= 5x – 20= 10 5x= 30 x= 6

26. Solve Exponential Equations Solve 3 x = 7. Round to the nearest hundredth. 3 x = 7 log 3 x = log 7 x log 3= log 7 x= or about 1.77 When the variable is the exponent, take the log/ln of both sides.

27. Solve Exponential Equations Solve e 2 x + 1 = 8. Give exact and round to the nearest hundredth. e 2x + 1 = 8 ln e 2x + 1 = ln 8 2x + 1= ln 8 x= or about 0.54

28. Solve log (3 x – 4) = 1 + log (2 x + 3). log (3x – 4)= 1 + log (2x + 3) Check for Extraneous Solutions log (3x – 4) – log (2x + 3)= 1 = 1

29. Check for Extraneous Solutions = 10 3x – 4= 10(2x + 3) 3x – 4= 20x + 30 –17x= 34 x= –2 Since neither log (–10) or log (–1) is defined, x = –2 is an extraneous solution. Answer: no solution

30. Solve and check. 4 x – 1 = 5 log 4 x – 1 = log 5 5 is not a power of 4, so take the log of both sides. (x – 1)log 4 = log 5 Apply the Power Property of Logarithms. Solving Exponential Equations Divide both sides by log 4. x = 1 + ≈ 2.161 log5 log4 x –1 = log5 log4 Exact and approximate answers

31. Solve. Solving Logarithmic Equations Write as a quotient. log 4 100 – log 4 ( x + 1) = 1 x = 24 Use 4 as the base for both sides. Use inverse properties on the left side. 100 x + 1 log 4 ( ) = 1 4 log 4 = 4 1 100 x + 1 ( ) = 4 100 x + 1

32. DOUBLING YOUR INVESTMENT. How long does it take for an investment to double at an annual interest rate of 8.5% compounded continuously? How long does it take for an investment to triple at an annual interest rate of 7.2% compounded continuously?