1. College Algebra Chapter 4 Exponential and Logarithmic Functions
Section 4.5
Exponential and Logarithmic Equations and Applications
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2. Concepts Solve Exponential Equations Solve Logarithmic Equations
Use Exponential and Logarithmic Equations in Applications

3. Concept 1 Solve Exponential Equations

4. Solve Exponential Equations
Some exponential equations can be solved using the equivalence property of exponential expressions: If b, x, and y are real numbers with b > 0 and b ≠ 1, then
Most exponential equations cannot be rewritten to have matching bases. To solve these equations, isolate the exponential expression, take a logarithm of both sides, and then apply the power property of logarithms.

5. Example 1
Solve
Solution:

6. Example 2
Solve
Solution:

7. Example 3
Solve
Solution:

8. Skill Practice 1
Solve.

9. Example 4
Solve
Solution:

10. Example 5
Solve
Solution:

11. Skill Practice 2
Solve.

12. Example 6
Solve
Solution:

13. Skill Practice 3
Solve.

14. Example 7
Solve
Solution:

15. Skill Practice 4
Solve.

16. Example 8

17. Skill Practice 5
Solve.

18. Concept 2 Solve Logarithmic Equations

19. Solve Logarithmic Equations (1 of 2)
Note the difference between these two equations:
In the first equation, all terms have a logarithm. The second equation is a mix of logarithmic and constant terms. These two equations require different styles of solution.

20. Solve Logarithmic Equations (2 of 2)
For solving equations of the first type, we will use the equivalence property of logarithmic expressions. If b, x, and y are real numbers with b > 0 and b ≠ 1, then
To solve equations that are a mix of logarithms and constants, collect all logarithms together, then rewrite the equation in exponential form.

21. Example 9
Solve
Solution:

22. Example 10
Solve log (x - 7) = log (5 - x) Solution: x – 7 = 5 - x 2x = 12 x = 6 log (5 - x) = log (5 - 6) = log (-1) undefined no solution

23. Skill Practice 6
Solve.

24. Example 11
Solve log (3x - 4) – log (x + 1) = log 2 Solution:

25. Example 12
Solve ln (x + 3) + ln x = ln 10 Solution:

26. Example 13
Solve
Solution:

27. Skill Practice 7
Solve. ln x + ln (x - 8) = ln (x - 20)

28. Example 14
Solve
Solution:

29. Example 15
Solve
Solution:

30. Example 16
Solve log (x + 15) = 2 – log x Solution: log (x + 15) = 2 – log x log (x + 15) + log x = 2 log x(x + 15) = 2 log base is 10

31. Example 17
Solve
Solution:

32. Skill Practice 8
Solve.

33. Skill Practice 9
Solve. log (t - 18) = 1.4

34. Skill Practice 10
Solve.

35. Concept 3 Use Exponential and Logarithmic Equations in Applications

36. Example 18
If $20,000 is invested in an account earning 2.5% interest compounded continuously, determine how long it will take the money to grow to $45,000. Round to the nearest year. Use the model
Solution: A = 45000, P = 20000, r = 0.025

37. Skill Practice 11
Determine how long it will take $8000 compounded monthly at 6% to double. Round to 1 decimal place.

38. Example 19 (1 of 2)
The following formula relates the energy E (in joules) released by an earthquake of magnitude M. log E = M (M > 5) What is the energy released by a magnitude 6 and a magnitude 7 earthquake? Solution:

39. Example 19 (2 of 2)
As a comparison, a 1 megaton nuclear weapon would have an energy release of
joules and the eruption of the volcano Krakatoa in 1883 produced an energy release of

40. Skill Practice 12
Find the intensity of sound from a leaf blower if the decibel level is 115 dB.
Is the intensity of sound from a leaf blower above the threshold for pain?