1. 3.4 – Solving Exponential and Logarithmic Equations Ch. 3 – Exponential and Logarithmic Functions

2. Properties of Logarithms Recall: a x = a y means x = y log a x = log a y means x = y log a a x = x a log a x = x Strategies for solving these equations: 1. Rewrite the equation in equivalent exponential/log form 2. Use inverse properties to cancel out logs and exponent bases Equivalent forms Inverse properties

3. Solve: 2 x = 32 Rewrite the 32 into an exponential phrase! 2 x = 2 5  x = 5 Solve: log x – log 12 = 0 log x = log 12  x = 12 Solve: ln x = -3 Use inverse properties! e ln x = e -3 x = e -3  x =.0498 These are the easy ones! They get harder!

4. Solve: 3(2 x ) = 42 Solve for x just like always… get x by itself! 2 x = 14 log 2 2 x = log 2 14 x = log 2 14 Use change of base to get x = 3.807 Solve: 6e 2x – 10 = 2 Get the e by itself first… 6e 2x = 12 e 2x = 2 ln e 2x = ln 2 2x = ln 2 x = (ln 2)/2  x =.347

5. Solve for x: 1..875 2. 11.02 3..883 4..380 5. 6.52

6. Solve for x: 1. 2.8 2..357 3. 90.597 4..640 5. 1.563

7. Solve for x. Round to the nearest tenth. 1. 1.7 2. 4.1 3. 6.9 4. 0.6 5. 0.8

8. Solve for x: 1..25 2..5 3. -.274 4. 1.5 5. 15.5

9. Solve for x. Round to the nearest tenth. 1. 1.8 2. 2.7 3. -3.3 4. 0.2 5. -6.1

10. Solve for x. Round to the nearest tenth. 1. -5.3 2. 0.7 3. -1.8 4. 2.4 5. No solution

11. Solve: e 2x – 3e x + 2 = 0 It’s like a quadratic, but with e’s, so write it as a quadratic! (e x ) 2 – 3e x + 2 = 0 Factor… (e x – 2)(e x – 1) = 0 Break into 2 equations!

12. Solve: ln (x – 2) + ln (2x – 3) = 2 ln x We can’t get rid of the ln’s until there are only 1 per side of the equation… …so we use our addition and exponent property of logs! ln ((x – 2)(2x – 3)) = ln x 2 Now we can lose the ln’s! (x – 2)(2x – 3) = x 2 2x 2 – 7x + 6 = x 2 x 2 – 7x + 6 = 0 Factor… (x – 6)(x – 1) = 0 x = 1, 6 WAIT!!! CHECK YOUR SOLUTIONS!!! x = 1 is extraneous, so final answer is x = 6

13. Write the exponential in logarithmic form. 1. log 3 4 = 64 2. log 3 64 = 4 3. log 4 3 = 64 4. log 64 4 = 3 5. log 4 64 = 3

14. Solve for x. Round to the nearest hundredth. 1. -1 2. 5.39 3. 0.52 4. 0 5. No Solution

15. Solve for x. Round to the nearest tenth. 1. 9 2. 8.5 3. -1, 9 4. 4.1 5. No Solution

16. Solve for x. Round to the nearest hundredth. 1. 0.08 2. -0.51 3. 0.76 4. -0.88 5. No Solution

17. Math Team Problem: Solve for w… 1. 25.3 2. 6 3. 8 4. 16 5. 85.3