Solving Logarithmic Equations

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Show your work for each problem.

1.
Which equation is the exponential form of $\log_4 x = 3$ ?
1. a) $x = 4^3$
2. b) $x = 3^4$
3. c) $x = \frac{4}{3}$
4. d) $x = 3 \cdot 4$
2.
Solve $\log_5 x = 3$ .
3.
Solve $\log_4(x - 3) = 2$ .
4.
Which value can NOT be a solution of an equation that contains $\log_3(x - 5)$ ?
1. a) $x = 32$
2. b) $x = 14$
3. c) $x = 4$
4. d) $x = 6$
5.
Solve $\log_2(3x - 1) = 3$ .
6.
Solve $\ln x = 2$ . Round to two decimal places.
7.
Solve $\log_3(5x - 4) = \log_3(2x + 11)$ .
8.
Solve $2 \log_5 x = \log_5 36$ .
9.
Which expression is the exact solution of $7^x = 50$ ?
1. a) $\log 50 - \log 7$
2. b) $\frac{\log 50}{\log 7}$
3. c) $\frac{50}{7}$
4. d) $\log \frac{50}{7}$
10.
Solve $5^x = 40$ by taking the common logarithm (base 10) of both sides. Round to two decimal places.
11.
Solve $2^x = 20$ . Round to two decimal places.
12.
Solve $\log_2(x + 6) - \log_2(x - 1) = 3$ .
13.
Solve $\log_6 x + \log_6(x + 1) = 1$ .
14.
Solve $\log_2 x + \log_2(x - 6) = 4$ .
1. a)only $x = 8$
2. b) $x = 8$ and $x = -2$
3. c)no solution
4. d)only $x = -2$
15.
Solve $\log_3 x + \log_3(x - 8) = 2$ .
16.
Solve $\log(x + 9) - \log x = 1$ , where $\log$ is the common logarithm (base 10).
17.
Solve $\log(x + 3) + \log x = 1$ (common logarithm, base 10).
1. a)only $x = 2$
2. b) $x = 2$ and $x = -5$
3. c)no solution
4. d)only $x = -5$
18.
Solve $4 \cdot 3^{x - 1} = 100$ . Round to two decimal places.