Power Rule for Logarithms

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Show your work for each problem.

1.
Which expression is equal to $\log_3(x^5)$ for $x > 0$ ?
1. a) $5 \log_3 x$
2. b) $(\log_3 x)^5$
3. c) $\log_3(5x)$
4. d) $\log_3 5 + \log_3 x$
2.
Evaluate $\log_2(8^4)$ .
3.
Evaluate $\log_3(9^4)$ using the power rule.
4.
Which expression is equal to $\log \sqrt{x}$ for $x > 0$ ?
1. a) $2 \log x$
2. b) $\log \frac{x}{2}$
3. c) $\sqrt{\log x}$
4. d) $\frac{1}{2} \log x$
5.
Evaluate $\log_5(25^3)$ .
6.
Evaluate $\log \sqrt{1000}$ (base 10).
7.
Write $3 \log x - 2 \log y$ as a single logarithm.
1. a) $\log(x^3 - y^2)$
2. b) $\frac{\log x^3}{\log y^2}$
3. c) $\log \frac{x^3}{y^2}$
4. d) $\log \frac{3x}{2y}$
8.
Evaluate $\log_4 \sqrt{64}$ . Give the answer as a fraction or a decimal.
9.
Expand $\log_2(x^3 y^5)$ as $a \log_2 x + b \log_2 y$ , then evaluate it for $x = 2$ and $y = 4$ .
10.
Which statement is true?
1. a) $3 \log 100 = 8$
2. b) $\log(100^3) = 6$
3. c) $\log(100^3) = 8$
4. d) $(\log 100)^3 = 6$
11.
If $\log_3 x = 2.5$ , what is $\log_3(x^4)$ ?
12.
Write $2 \log 5 + \log 4$ as a single logarithm and evaluate it.
13.
Expand $\ln \dfrac{x^4}{\sqrt{y}}$ as $a \ln x + c \ln y$ , then evaluate it when $\ln x = 1$ and $\ln y = 6$ .
14.
Expand $\ln \dfrac{x^2 \sqrt[3]{y}}{z^4}$ (all variables positive).
1. a) $2 \ln x + \frac{1}{3} \ln y - 4 \ln z$
2. b) $2 \ln x + \frac{1}{3} \ln y + 4 \ln z$
3. c) $2 \ln x + 3 \ln y - 4 \ln z$
4. d) $\frac{2}{3} \ln(xy) - 4 \ln z$
15.
Use $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$ to approximate $\log 48$ . Give 3 decimal places.
16.
Evaluate $3 \log_2 4 - \frac{1}{2} \log_2 16$ by condensing it into one logarithm.
17.
Use $\ln 2 \approx 0.693$ and $\ln 3 \approx 1.099$ to approximate $\ln \sqrt{72}$ . Work only with these values and do not round.
18.
How many digits does $3^{50}$ have? Use $\log 3 \approx 0.477$ .