1. 3.4 Quick Review Express In 56 in terms of ln 2 and ln 7.
A. ln 23 + ln 7
B. ln 7 – 3 ln 2
C. 3 ln 2 – ln 7
D. 3 ln 2 + ln 7

2. 3.4 Quick Review Express In 56 in terms of ln 2 and ln 7.
A. ln 23 + ln 7
B. ln 7 – 3 ln 2
C. 3 ln 2 – ln 7
D. 3 ln 2 + ln 7

3. 3.4 Quick Review
Expand A. B. C. D.

4. 3.4 Quick Review
Expand A. B. C. D.

5. 3.4 Quick Review
Condense
A. AAAA
B. BBBB
C. CCCC
D. DDDD

6. 3.4 Quick Review
Condense
A. AAAA
B. BBBB
C. CCCC
D. DDDD

7. 3.4 Quick Review Estimate log3 7 to the nearest whole number. A. 0
D. 3

8. 3.4 Quick Review Estimate log3 7 to the nearest whole number. A. 0
D. 3

9. Key Concept 1

10. 4x + 2 = 16x – 3 Original equation 4x + 2 = (42)x – 3 42 = 16
Solve Exponential Equations Using One-to-One Property
A. Solve 4x + 2 = 16x – 3.
4x + 2 = 16x – 3 Original equation
4x + 2 = (42)x – 3 42 = 16
4x + 2 = 42x – 6 Power of a Power
x + 2 = 2x – 6 One-to-One Property
2 = x – 6 Subtract x from each side.
8 = x Add 6 to each side.
Answer:
Example 1

11. B. Solve . Original equation Power of a Power
Solve Exponential Equations Using One-to-One Property
B. Solve
Original equation
Power of a Power
Example 1

12. One-to-One Property. Answer:
Solve Exponential Equations Using One-to-One Property
One-to-One Property.
Answer:
Example 1

13. Solve 25x + 2 = 54x.
A. 1 B.
C. 2
D. –2
Example 1

14. Solve 25x + 2 = 54x.
A. 1 B.
C. 2
D. –2
Example 1

15. A. Solve 2 ln x = 18. Round to the nearest hundredth.
Solve Logarithmic Equations Using One-to-One Property
A. Solve 2 ln x = 18. Round to the nearest hundredth.
Method Use exponentiation.
2 ln x = 18 Original equation
ln x = 9 Divide each side by 2.
eln x = e9 Exponentiate each side.
x = e9 Inverse Property
x ≈ Use a calculator.
Example 2

16. Method 2 Write in exponential form. 2 ln x = 18 Original equation
Solve Logarithmic Equations Using One-to-One Property
Method Write in exponential form.
2 ln x = 18 Original equation
ln x = 9 Divide each side by 2.
x = e9 Write in exponential form.
x ≈ Use a calculator.
Answer:
Example 2

17. B. Solve 7 – 3 log 10x = 13. Round to the nearest hundredth.
Solve Logarithmic Equations Using One-to-One Property
B. Solve 7 – 3 log 10x = 13. Round to the nearest hundredth.
7 – 3 log 10x = 13 Original equation
–3 log 10x = 6 Subtract 7 from each side.
log 10x = –2 Divide each side by –3.
10–2 =10x Write in exponential form.
10–3 = x Divide each side by 10.
= x = 10–3.
Answer:
Example 2

18. C. Solve log5 x 4 = 20. Round to the nearest hundredth.
Solve Logarithmic Equations Using One-to-One Property
C. Solve log5 x 4 = 20. Round to the nearest hundredth.
log5x4 = 20 Original equation
4 log5x = 20 Power Property
log5x = 5 Divide each side by 4.
x = 55 Write in exponential form
x = Simplify.
Answer:
Example 2

19. Solve 2 log2x 3 = 18.
A. 81
B. 27
C. 9
D. 8
Example 2

20. Solve 2 log2x 3 = 18.
A. 81
B. 27
C. 9
D. 8
Example 2

21. Key Concept 2

22. A. Solve log2 5 = log2 10 – log2 (x – 4).
Solve Exponential Equations Using One-to-One Property
A. Solve log2 5 = log2 10 – log2 (x – 4).
log25 = log210 – log2(x – 4) Original equation
log25 = Quotient Property
5 = One-to-One Property
5x – 20 = 10 Multiply each side by x – 4.5
5x = 30 Add 20 to each side.
x = 6 Divide each side by 5.
Answer:
Example 3

23. log5(x2 + x) = log520 Original equation
Solve Exponential Equations Using One-to-One Property
B. Solve log5 (x2 + x) = log5 20.
log5(x2 + x) = log520 Original equation
x2 + x = 20 One-to-One Property
x2 + x – 20 = 0 Subtract 20 from each side.
(x – 4)(x + 5) = 0 Factor x 2 + x – 20 into linear factors.
x = –5 or 4 Solve for x. Check this solution.
Answer:
Example 3

24. Solve log315 = log3x + log3(x – 2).
A. 5
B. –3
C. –3, 5
D. no solution
Example 3

25. Solve log315 = log3x + log3(x – 2).
A. 5
B. –3
C. –3, 5
D. no solution
Example 3

26. A. Solve 3x = 7. Round to the nearest hundredth.
Solve Exponential Equations
A. Solve 3x = 7. Round to the nearest hundredth.
3x = 7 Original equation
log 3x = log 7 Take the common logarithm of each side.
x log 3 = log 7 Power Property
x = or about 1.77 Divide each side by log 3 and use a calculator.
Answer:
Example 4

27. B. Solve e2x + 1 = 8. Round to the nearest hundredth.
Solve Exponential Equations
B. Solve e2x + 1 = 8. Round to the nearest hundredth.
e2x + 1 = 8 Original equation
ln e2x + 1 = ln 8 Take the natural logarithm of each side.
2x + 1 = ln 8 Inverse Property
x = or about 0.54 Solve for x and use a calculator.
Answer:
Example 4

28. Solve 4x = 9. Round to the nearest hundredth.
B. 1.58
C. 2.25
D. 0.44
Example 4

29. Solve 4x = 9. Round to the nearest hundredth.
B. 1.58
C. 2.25
D. 0.44
Example 4

30. Solve 36x – 3 = 24 – 4x. Round to the nearest hundredth.
Solve in Logarithmic Terms
Solve 36x – 3 = 24 – 4x. Round to the nearest hundredth.
36x – 3 = 24 – 4x Original equation
ln 36x – 3 = ln 24 – 4x Take the natural logarithm of each side.
(6x – 3) ln 3 = (4 – 4x) ln 2 Power Property
6x ln 3 – 3 ln3 = 4 ln 2 – 4x ln 2 Distributive Property
6x ln 3 + 4x ln 2 = 4 ln ln3 Isolate the variable on the left side of the equation.
x(6 ln ln 2) = 4 ln ln3 Distributive Property
Example 5

31. x(ln 36 + ln 24) = ln 24 + ln 33 Power Property
Solve in Logarithmic Terms
x(ln 36 + ln 24) = ln 24 + ln 33 Power Property
x ln [36(24)] = ln [24(33)] Product Property
x ln 11,664 = ln (24) = 11,664 and 24(33) = 432
x = Divide each side by ln 11,664.
x ≈ 0.65 Use a calculator.
Answer:
Example 5

32. Solve 4x + 2 = 32 – x. Round to the nearest hundredth.
B. 1.08
C. 0.68
D. –0.23
Example 5

33. Solve 4x + 2 = 32 – x. Round to the nearest hundredth.
B. 1.08
C. 0.68
D. –0.23
Example 5

34. e2x – ex – 2 = 0 Original equation
Solve Exponential Equations in Quadratic Form
Solve e2x – ex – 2 = 0.
e2x – ex – 2 = 0 Original equation
u2 – u – 2 = 0 Write in quadratic form by letting u = ex.
(u – 2)(u + 1) = 0 Factor.
u = 2 or u = –1 Zero Product Property
ex = 2 ex = –1 Replace u with ex.
Example 6

35. ln ex = ln 2 ln ex = ln (–1) Take the natural logarithm of each side.
Solve Exponential Equations in Quadratic Form
ln ex = ln 2 ln ex = ln (–1) Take the natural logarithm of each side.
x = ln 2 x = ln (–1) Inverse Property or about 0.69
The only solution is x = ln 2 because ln (–1) is extraneous.
Answer:
Example 6

36. e2x – ex – 2 = 0 Original equation
Solve Exponential Equations in Quadratic Form
Check
e2x – ex – 2 = 0 Original equation
e2(ln 2) – eln 2 – 2 = 0 Replace x with ln 2.
eln 22– eln 2 – 2 = 0 Power Property
22 – 2 – 2 = 0 Inverse Property
0 = 0 Simplify.
Example 6

37. Solve e2x + ex – 12 = 0. A. ln 3 B. ln 3, ln 4 C. ln 4
D. ln 3, ln (–4)
Example 6

38. Solve e2x + ex – 12 = 0. A. ln 3 B. ln 3, ln 4 C. ln 4
D. ln 3, ln (–4)
Example 6

39. Solve log x + log (x – 3) = log 28.
Solve Logarithmic Equations
Solve log x + log (x – 3) = log 28.
log x + log (x – 3) = log 28 Original equation
log x(x – 3) = log 28 Product Property
log (x 2 – 3x) = log 28 Simplify.
x 2 – 3x = 28 One-to-One Property
x 2 – 3x – 28 = 0 Subtract 28 from each side.
(x – 7)(x + 4) = 0 Factor.
x = 7 or x = – 4 Zero Product Property
Example 7

40. The only solution is x = 7 because –4 is an extraneous solution.
Solve Logarithmic Equations
The only solution is x = 7 because –4 is an extraneous solution.
Answer:
Example 7

41. Solve ln x + ln (5 – x) = ln 6.
A. 2
B. 3
C. 2, 3
D. –2, –3
Example 7

42. Solve ln x + ln (5 – x) = ln 6.
A. 2
B. 3
C. 2, 3
D. –2, –3
Example 7

43. Solve log (3x – 4) = 1 + log (2x + 3).
Check for Extraneous Solutions
Solve log (3x – 4) = 1 + log (2x + 3).
log (3x – 4) = 1 + log (2x + 3) Original Equation
log (3x – 4) – log (2x + 3) = 1 Subtract log (2x + 3) from each side.
= 1 Quotient Property
= log 101 Inverse Property
= log = 10
Example 8

44. 3x – 4 = 10(2x + 3) Multiply each side by 2x + 3.
Check for Extraneous Solutions
= 10 One-to-One Property
3x – 4 = 10(2x + 3) Multiply each side by 2x + 3.
3x – 4 = 20x + 30 Distributive Property
–17x = 34 Subtract 20x and add 4 to each side.
x = –2 Divide each side by –17.
Example 8

45. Section 3.4 Homework
p. 196; 5-13 odd, 23, 24, 27, 33, 35, 39, 41, 51-53, 61-63, 71