1. Topic 10 : Exponential and Logarithmic Functions 10.1.1 Solving Exponential and Logarithmic Equations

2. Previously we studied…

3. Now we will study…

4. Exponential Function The function defined by is called an exponential function with base b and exponent x. The domain of f is the set of all real numbers.

5. Identify Exponential Functions Which of the following are exponential functions? y = 3 x y = x 3 y = 2(7) x y = 2(-7) x yes no

6. Identify the Base Identify the base in each of the following. y = 3 x y = 2(7) x y = 3a x y = 4 x - 3

7. Example The exponential function with base 2 is the function with domain (– ,  ). The values of f(x) for selected values of x follow:

8. More Examples The values of f(x) for selected values of x follow:

9. Recall Laws of Exponents Let a and b be positive numbers and let x and y be real numbers. Then, 1. 2. 3. 4. 5.

10. Evaluate Exponential Functions y = 3 x for x = 4 y = 2(7) x for x = 3 y = -2(4 x ) for x = 3/2 y = 81 y = 686 y = -16

11. One way to solve exponential equations is to use the property that if 2 powers w/ the same base are equal, then their exponents are equal. For b>0 & b≠1 if b x = b y, then x=y Exponential Equations

12. Solve by equating exponents 4 3x = 8 x+1 (2 2 ) 3x = (2 3 ) x+1 rewrite w/ same base 2 6x = 2 3x+3 6x = 3x+3 x = 1 Check → 4 3*1 = 8 1+1 64 = 64

13. Your turn! 2 4x = 32 x-1 2 4x = (2 5 ) x-1 4x = 5x-5 5 = x Be sure to check your answer!!!

14. But… what happens when the base are different? When you can’t rewrite using the same base, you can solve by taking a log of both sides

15. Exponential vs Logarithms We’ve discussed exponential equations of the form y = b x (b > 0, b ≠ 1) You may recall that y is called the logarithm of x to the base b, and is denoted log b x. –Logarithm of x to the base b y = log b x if and only if x = b y (x > 0)

17. Rewriting Logarithmic Equations:

18. Examples Solve log 3 x = 4 for x: Solution By definition, log 3 x = 4 implies x = 3 4 x = 81

19. Examples Solve log 16 4 = x for x: Solution log 16 4 = x is equivalent to 4 = 16 x 4 = (4 2 ) x or 4 1 = 4 2x from which we deduce that

20. Examples

21. Logarithmic Notation

22. Laws of Logarithms If m and n are positive numbers, then 1. 2. 3. 4. 5.

23. Change base Formula

24. When you can’t rewrite using the same base, you can solve by taking a log of both sides 2 x = 7 log 2 x = log 7 x log 2 = log 7 x = ≈ 2.807

25. Practice…Practice…Practice