1. 8-5 Exponential and Logarithmic Equations Solving logarithmic & exponential equations

2. Objectives Solving Exponential Equations Solving Logarithmic Equations

3. Vocabulary An equation of the form b cx = a is and exponential equation. If m = n, then log m = log n

4. Solve log 5 2x = 16. 5 2x = 16 5 2x = log 16Take the common logarithm of each side. 2x log 5 = log 16Use the power property of logarithms. x = Divide each side by 2 log 5. log 16 2 log 5 0.8614Use a calculator. Check: 5 2x 16 5 2(0.8614) 16 Solving an Exponential Equation

5. Solve 4 3x = 1100 by graphing. The solution is x 1.684 Graph the equations y = 4 3x and y = 1100. Find the point of intersection. Solving an Exponential Equation by Graphing

6. 4.589 Use a calculator. 1.387 log 3 log xMultiply each side by log 3. 1.387 0.4771 log x Use a calculator. 0.6617 log xSimplify. x 10 0.6617 Write in exponential form. The expression log 6 12 is approximately equal to 1.3869, or log 3 4.589. (continued) Continued

7. Solve 5 2x = 120 using tables. Enter y 1 = 5 2x – 120. Use tabular zoom-in to find the sign change, as shown at the right. The solution is x  1.487. Solving an Exponential Equation by Tables

8. The population of trout in a certain stretch of the Platte River is shown for five consecutive years in the table, where 0 represents the year 1997. If the decay rate remains constant, in the beginning of which year might at most 100 trout remain in this stretch of river? Time t01234 Pop. P(t)50004000320125612049 Step 1: Enter the data into your calculator. Step 2: Use the Exp Reg feature to find the exponential function that fits the data. Real-World Example

9. Step 3: Graph the function and the line y = 100. Step 4: Find the point of intersection. The solution is x 18, so there may be only 100 trout remaining in the beginning of the year 2015. (continued) Continued

10. Vocabulary For any positive numbers, M, b, c, with b ≠ 1 and c ≠ 1, log b M = log c M log c b

11. Use the Change of Base Formula to evaluate log 6 12. Then convert log 6 12 to a logarithm in base 3. log 6 12 = Use the Change of Base Formula. log 12 log 6 1.387Use a calculator. 1.0792 0.7782 log 6 12 = log 3 xWrite an equation. 1.387 log 3 xSubstitute log 6 12 = 1.3868 1.387Use the Change of Base Formula. log x log 3 Using the Change of Base Formula

12. Solve log (2x – 2) = 4. log (2x – 2) = 4 2x – 2 = 10 4 Write in exponential form. 2x – 2 = 10000 x = 5001Solve for x. log 10 4 = 4 log 10,000 4 log (2 5001 – 2) 4 Check:log (2x – 2) 4 Solving a Logarithmic Equation

13. Solve 3 log x – log 2 = 5. 3 log x – log 2 = 5 x32x32 Log ( ) = 5Write as a single logarithm. x32x32 = 10 5 Write in exponential form. x 3 = 2(100,000)Multiply each side by 2. x = 10 200, or about 58.48. 3 The solution is 10 200, or about 58.48. 3 Using Logarithmic Properties to Solve an Equation

14. Homework 8-5 Pg 464 # 1, 2, 13, 14, 23, 25, 26, 33, 34, 42, 43