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5.5 Solving Exponential Equations and Logarithmic Equations

Solve exponential equations.
Solve logarithmic equations.

Solving Exponential Equations

Equations with variables in the exponents, such as

$3^{x} = 20 and 2^{5 x} = 64,$

are called exponential equations.

Sometimes, as is the case with the equation $2^{5 x} = 64$ , we can write each side as a power of the same number:

$2^{5 x} = 2^{6} .$

We can then set the exponents equal and solve:

$\begin{array}{rcl} 5 x & = & 6 \\ x & = & \frac{6}{5}, or 1.2. \end{array}$

We use the following property to solve exponential equations.

This property follows from the fact that for any $a > 0$ , $a \neq 1$ , $f (x) = a^{x}$ is a one-to-one function. If $a^{x} = a^{y}$ , then $f (x) = f (y)$ . Then since f is one-to-one, it follows that $x = y$ . Conversely, if $x = y$ , it follows that $a^{x} = a^{y}$ , since we are raising a to the same power in each case.

Example 1

Solve: $2^{3 x - 7} = 32$ .

Now Try Exercise 7.

Another property that is used when solving some exponential equations and logarithmic equations is as follows.

This property follows from the fact that for any $a > 0$ , $a \neq 1$ , $f (x) = \log_{a} x$ is a one-to-one function. If $\log_{a} x = \log_{a} y$ , then $f (x) = f (y)$ . Then since f is one-to-one, it follows that $x = y$ . Conversely, if $x = y$ , it follows that $\log_{a} x = \log_{a} y$ , since we are taking the logarithm of the same number in each case.

When it does not seem possible to write each side as a power of the same base, we can use the property of logarithmic equality and take the logarithm with any base on each side and then use the power rule for logarithms.

Example 2

Solve: $3^{x} = 20$ .

Now Try Exercise 11.

In Example 2, we took the common logarithm on both sides of the equation. Any base will give the same result. Let’s try base 3. We have

$\begin{array}{rcll} 3^{x} & = & 20 \\ \log_{3} 3^{x} & = & \log_{3} 20 \\ x & = & \log_{3} 20 & \log_{a} a^{x} = x \\ x & = & \frac{\log 20}{\log 3} & Using the change-of-base formula \\ x & \approx & 2.7268. \end{array}$

Note that we must change the base in order to do the final calculation.

Example 3

Solve: $100 e^{0.08 t} = 2500$ .

Now Try Exercise 19.

Example 4

Solve: $4^{x + 3} = 3^{- x}$ .

Now Try Exercise 21.

Example 5

Solve: $e^{x} + e^{- x} - 6 = 0$ .

Now Try Exercise 25.

Solving Logarithmic Equations

Equations containing variables in logarithmic expressions, such as $\log_{2} x = 4$ and $\log x + \log (x + 3) = 1$ , are called logarithmic equations. To solve logarithmic equations algebraically, we first try to obtain a single logarithmic expression on one side and then write an equivalent exponential equation.

Example 6

Solve: $\log_{3_{}} x = - 2$ .

Now Try Exercise 33.

Example 7

Solve: $\log x + \log (x + 3) = 1$ .

Now Try Exercise 41.

Example 8

Solve: $\log_{3} (2 x - 1) - \log_{3} (x - 4) = 2$ .

Now Try Exercise 45.

Example 9

Solve: $\ln (4 x + 6) - \ln (x + 5) = \ln x$ .

Now Try Exercise 43.

5.5 Exercise Set

Solve the exponential equation.

1. $3^{x} = 81$
2. $2^{x} = 32$
3. $2^{2 x} = 8$
4. $3^{7 x} = 27$
5. $2^{x} = 33$
6. $2^{x} = 40$
7. $5^{4 x - 7} = 125$
8. $4^{3 x - 5} = 16$
9. $27 = 3^{5 x} \cdot 9^{x^{2}}$
10. $3^{x^{2} + 4 x} = \frac{1}{27}$
11. $84^{x} = 70$
12. $28^{x} = 10^{- 3 x}$
13. $10^{- x} = 5^{2 x}$
14. $15^{x} = 30$
15. $e^{- c} = 5^{2 c}$
16. $e^{4 t} = 200$
17. $e^{t} = 1000$
18. $e^{- t} = 0.04$
19. $e^{- 0.03 t} = 0.08$
20. $1000 e^{0.09 t} = 5000$
21. $3^{x} = 2^{x - 1}$
22. $5^{x + 2} = 4^{1 - x}$
23. ${(3.9)}^{x} = 48$
24. $250 - {(1.87)}^{x} = 0$
25. $e^{x} + e^{- x} = 5$
26. $e^{x} - 6 e^{- x} = 1$
27. $3^{2 x - 1} = 5^{x}$
28. $2^{x + 1} = 5^{2 x}$
29. $2 e^{x} = 5 - e^{- x}$
30. $e^{x} + e^{- x} = 4$

Solve the logarithmic equation.

31. $\log_{5} x = 4$
32. $\log_{2} x = - 3$
33. $\log x = - 4$
34. $\log x = 1$
35. $\ln x = 1$
36. $\ln x = - 2$
37. $\log_{64} \frac{1}{4} = x$
38. $\log_{125} \frac{1}{25} = x$
39. $\log_{2} (10 + 3 x) = 5$
40. $\log_{5} (8 - 7 x) = 3$
41. $\log x + \log (x - 9) = 1$
42. $\log_{2} (x + 1) + \log_{2} (x - 1) = 3$
43. $\log_{2} (x + 20) - \log_{2} (x + 2) = \log_{2} x$
44. $\log (x + 5) - \log (x - 3) = \log 2$
45. $\log_{8} (x + 1) - \log_{8} x = 2$
46. $\log x - \log (x + 3) = - 1$
47. $\log x + \log (x + 4) = \log 12$
48. $\log_{3_{}} (x + 14) - \log_{3_{}} (x + 6) = \log_{3} x$
49. $\log (x + 8) - \log (x + 1) = \log 6$
50. $\ln x - \ln (x - 4) = \ln 3$
51. $\log_{4} (x + 3) + \log_{4} (x - 3) = 2$
52. $\ln (x + 1) - \ln x = \ln 4$
53. $\log (2 x + 1) - \log (x - 2) = 1$
54. $\log_{5} (x + 4) + \log_{5} (x - 4) = 2$
55. $\ln (x + 8) + \ln (x - 1) = 2 \ln x$
56. $\log_{3} x + \log_{3} (x + 1) = \log_{3} 2 + \log_{3} (x + 3)$

Solve.

57. $\log_{6} x = 1 - \log_{6} (x - 5)$
58. $2^{x^{2} - 9 x} = \frac{1}{256}$
59. $9^{x - 1} = 100 (3^{x})$
60. $2 \ln x - \ln 5 = \ln (x + 10)$
61. $e^{x} - 2 = - e^{- x}$
62. $2 log 50 = 3 log 25 + \log (x - 2)$

Skill Maintenance

In Exercises 63–66:

Find the vertex.
Find the axis of symmetry.
Determine whether there is a maximum or a minimum value and find that value. [3.3]

63. $g (x) = x^{2} - 6$
64. $f (x) = - x^{2} + 6 x - 8$
65. $G (x) = - 2 x^{2} - 4 x - 7$
66. $H (x) = 3 x^{2} - 12 x + 16$

Synthesis

Solve using any method.

67. $\frac{e^{x} + e^{- x}}{e^{x} - e^{- x}} = 3$
68. $\ln (\ln x) = 2$
69. $\sqrt{\ln x} = \ln \sqrt{x}$
70. $\ln \sqrt[4]{x} = \sqrt{\ln x}$
71. ${(\log_{3} x)}^{2} - \log_{3} x^{2} = 3$
72. $\log_{3} (\log_{4} x) = 0$
73. $\ln x^{2} = {(\ln x)}^{2}$
74. $x (\ln \frac{1}{6}) = \ln 6$
75. $5^{2 x} - 3 \cdot 5^{x} + 2 = 0$
76. $x^{\log x} = \frac{x^{3}}{100}$
77. $\ln x^{\ln x} = 4$
78. $| 2^{x^{2}} - 8 | = 3$
79. $\frac{\sqrt{{(e^{2 x} \cdot e^{- 5 x})}^{- 4}}}{e^{x} \div e^{- x}} = e^{7}$
80. Given that $a = {(\log_{125} 5)}^{\log_{5} 125}$ , find the value of $\log_{3} a$ .
81. Given that $a = \log_{8} 225$ and $b = \log_{2} 15$ , express a as a function of b.
82. Given that $f (x) = e^{x} - e^{- x}$ , find $f^{- 1} (x)$ if it exists.

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Table of Contents for 5.5 Solving Exponential Equations and Logarithmic Equations

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Table of Contents for
5.5 Solving Exponential Equations and Logarithmic Equations