1. Expand the following log Log 3 (x 2 yz / w 9 ) Condense the following log Log 2 (x) + log 2 (y) - 3log 2 (z)

5.  Essential Question: How do you solve exponential and logarithmic equations?  Standard: MM4A4. Students will investigate functions. a. Compare and contrast properties of functions within and across the following types: linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, and piecewise.

6. Exponential Equations

7. An exponential equation is one in which the variable occurs in the exponent. For example, 2 x = 7

8. Solving Exponential Equations The variable x presents a difficulty because it is in the exponent. To deal with this difficulty, we take the logarithm of each side and then use the Laws of Logarithms to “bring down x” from the exponent.

9. Solving Exponential Equations The power rule says that: log a A C = C log a A

10. Guidelines for Solving Exponential Equations 1.Isolate the exponential expression on one side of the equation. 2.Take the logarithm of each side, and then use the Laws of Logarithms to “bring down the exponent.” 3.Solve for the variable.

11. E.g. 1—Solving an Exponential Equation Find the solution of 3 x + 2 = 7 correct to six decimal places. We take the common logarithm of each side and use the power rule.

12. E.g. 1—Solving an Exponential Equation

13. E.g. 2—Solving an Exponential Equation Solve the equation 8e 2x = 20. We first divide by 8 to isolate the exponential term on one side.

14. E.g. 3—Solving Algebraically and Graphically Solve the equation e 3 – 2x = 4 algebraically and graphically

15. E.g. 3—Solving Algebraically The base of the exponential term is e. So, we use natural logarithms to solve. You should check that this satisfies the original equation. Solution 1

16. E.g. 4—Exponential Equation of Quadratic Type Solve the equation e 2x – e x – 6 = 0. To isolate the exponential term, we factor.

17. E.g. 4—Exponential Equation of Quadratic Type The equation e x = 3 leads to x = ln 3. However, the equation e x = –2 has no solution because e x > 0 for all x. Thus, x = l n 3 ≈ 1.0986 is the only solution. You should check that this satisfies the original equation.

18. Logarithmic Equations

19. A logarithmic equation is one in which a logarithm of the variable occurs. For example, log 2 (x + 2) = 5

20. Solving Logarithmic Equations To solve for x, we write the equation in exponential form. x + 2 = 2 5 x = 32 – 2 = 30

21. Solving Logarithmic Equations Another way of looking at the first step is to raise the base, 2, to each side. 2 log 2 (x + 2) = 2 5 x + 2 = 2 5 x = 32 – 2 = 30 The method used to solve this simple problem is typical.

22. Guidelines for Solving Logarithmic Equations 1.Isolate the logarithmic term on one side of the equation. You may first need to combine the logarithmic terms. 2.Write the equation in exponential form (or raise the base to each side). 3.Solve for the variable.

23. E.g. 6—Solving Logarithmic Equations Solve each equation for x. (a)ln x = 8 (b)log 2 (25 – x) = 3

24. E.g. 6—Solving Logarithmic Eqns. ln x = 8 x = e 8 Therefore, x = e 8 ≈ 2981. We can also solve this problem another way: Example (a)

25. E.g. 6—Solving Logarithmic Eqns. The first step is to rewrite the equation in exponential form. Example (b)

26. E.g. 7—Solving a Logarithmic Equation Solve the equation 4 + 3 log(2x) = 16 We first isolate the logarithmic term. This allows us to write the equation in exponential form.

27. E.g. 7—Solving a Logarithmic Equation

28.  Examples: Solve the following: e x = 323(2) x = 42

29. More examples Solve 4e 2x -3 = 2 2 x = 512

30. Solve the following  Ln(5) – ln(x) = 0  Ln(x) = -8

31.  2(3 2t -5 ) -4 = 11

32. e 2x -3e x + 2 = 0

33.  Ln(3x) = 2  Log 3 (5x-1) = Log 3 (x+7)

34. Solve (1/2) x = 32

35.  P 221 # 1-8, 17, 22, 27, 29-34, 39-44,85,96, 91,92