1. 8.5 – Exponential and Logarithmic Equations

2. CHANGE OF BASE FORMULA where M, b, and c are positive numbers and b, c do not equal one. Ex: Rewrite log 5 15 using the change of base formula

3. Steps for solving exponential equations  Take a common logarithm of each side  Use the power property of logarithms  Solve for x by dividing  Use a calculator to find the approximate value

4. Solving Exponential Equations 1. Take the log of both sides 2. Use the power property 3. Solve for x. Solve. Round to the nearest ten-thousandth. X=1.26194. Use a calculator. Check your answer – 3 1.2619 =4

5. Another Example 1. Take the log of both sides 2. Use the power property 3. Solve for x. Solve. Round to the nearest ten-thousandth. X=4.2009 – 4 = 0.20094. Use a calculator. Check your answer – 3 0.2009+4 =101

6. Let’s try some

10. CHANGE OF BASE – HOW IT WORKS  Use the change of base formula to evaluate. Then convert it to a logarithm of base 2. 1. Rewrite using the change of base formula 2. Use a calculator 3. Write an equation to convert to base 2

11. CHANGE OF BASE – HOW IT WORKS 6. Multiply both sides of the equation by log2 7. Use a calculator; simplify. 8. Write in exponential form. 5. Rewrite using the change of base formula 4. Substitute log 3 15=2.4650 X=5.52089. Use a calculator. Log 3 15 is approximately equal to 2.4650 or log 2 5.5208

12. Let’s try one  Use the change of base formula to evaluate. Then convert it to a logarithm of base 8. 1. Rewrite using the change of base formula 2. Use a calculator 3. Write an equation to convert to base 2

13. 6. Multiply both sides of the equation by log8 7. Use a calculator; simplify. 8. Write in exponential form. 5. Rewrite using the change of base formula 4. Substitute log 5 400=3.727 X=23019. Use a calculator. Log 5 400 is approximately equal to 3.7227 or log 8 2301

14. SOLVING SIMPLE LOG EQUATIONS 1. Use the product property 2. Write in exponential form. 3. Simplify 4. Solve for x.

15. Let’s try some

19. Solving exponential equations with a graphing calculator 1.Type two equations into y= Solution: 2.0408 2. Graph. Suggest Zoom fit (0) especially for large values 3. Use the calc function to find the intersection of the two graphs.