Power Rule for Logarithms

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) $\log_3 5 + \log_3 x$
Answer: $5 \log_3 x$
By the power rule, $\log_3(x^5) = 5 \log_3 x$ . The expression $(\log_3 x)^5$ raises the whole logarithm to the fifth power, which is different.
2.
Evaluate $\log_2(8^4)$ .
Answer: 12
$\log_2(8^4) = 4 \log_2 8 = 4 \cdot 3 = 12$ . Check: $8^4 = 2^{12} = 4096$ .
3.
Evaluate $\log_3(9^4)$ using the power rule.
Answer: 8
1. The power rule gives $4 \log_3 9$ . What is $\log_3 9$ ? 2
2. Multiply: $4 \cdot \log_3 9 = ?$ 8
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) $\frac{1}{2} \log x$
Answer: $\frac{1}{2} \log x$
$\log \sqrt{x} = \log x^{1/2} = \frac{1}{2} \log x$ . A square root halves the logarithm, it does not double it.
5.
Evaluate $\log_5(25^3)$ .
Answer: 6
$\log_5(25^3) = 3 \log_5 25 = 3 \cdot 2 = 6$ .
6.
Evaluate $\log \sqrt{1000}$ (base 10).
Answer: 3/2
1. Write $\sqrt{1000} = 1000^k$ . What is $k$ ? 1/2
2. What is $\log 1000$ ? 3
3. Power rule: multiply $k \cdot \log 1000$ . 3/2
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) $\log \frac{3x}{2y}$
Answer: $\log \frac{x^3}{y^2}$
$3 \log x = \log x^3$ and $2 \log y = \log y^2$ , so $3 \log x - 2 \log y = \log x^3 - \log y^2 = \log \frac{x^3}{y^2}$ .
8.
Evaluate $\log_4 \sqrt{64}$ . Give the answer as a fraction or a decimal.
Answer: 3/2
$\log_4 \sqrt{64} = \frac{1}{2} \log_4 64 = \frac{1}{2} \cdot 3 = \frac{3}{2} = 1.5$ . Check: $\sqrt{64} = 8$ and $4^{1.5} = 8$ .
9.
Expand $\log_2(x^3 y^5)$ as $a \log_2 x + b \log_2 y$ , then evaluate it for $x = 2$ and $y = 4$ .
Answer: 13
1. What is the coefficient $a$ of $\log_2 x$ ? 3
2. What is the coefficient $b$ of $\log_2 y$ ? 5
3. With $\log_2 2 = 1$ and $\log_2 4 = 2$ , compute $3 \cdot 1 + 5 \cdot 2$ . 13
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) $(\log 100)^3 = 6$
Answer: $\log(100^3) = 6$
$\log(100^3) = 3 \log 100 = 3 \cdot 2 = 6$ by the power rule. In contrast, $(\log 100)^3 = 2^3 = 8$ , because there the whole logarithm is cubed.
11.
If $\log_3 x = 2.5$ , what is $\log_3(x^4)$ ?
Answer: 10
$\log_3(x^4) = 4 \log_3 x = 4 \cdot 2.5 = 10$ . You never need to know $x$ itself.
12.
Write $2 \log 5 + \log 4$ as a single logarithm and evaluate it.
Answer: 2
1. Power rule backwards: $2 \log 5 = \log ?$ 25
2. Product rule: $\log 25 + \log 4 = \log ?$ 100
3. What is $\log 100$ ? 2
13.
Expand $\ln \dfrac{x^4}{\sqrt{y}}$ as $a \ln x + c \ln y$ , then evaluate it when $\ln x = 1$ and $\ln y = 6$ .
Answer: 1
1. What is the coefficient $a$ of $\ln x$ ? 4
2. What is the coefficient $c$ of $\ln y$ ? (Careful with the sign.) -1/2
3. Compute $4 \cdot 1 - \frac{1}{2} \cdot 6$ . 1
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) $\frac{2}{3} \ln(xy) - 4 \ln z$
Answer: $2 \ln x + \frac{1}{3} \ln y - 4 \ln z$
Quotient rule, product rule, then power rule: $\ln x^2 + \ln y^{1/3} - \ln z^4 = 2 \ln x + \frac{1}{3} \ln y - 4 \ln z$ .
15.
Use $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$ to approximate $\log 48$ . Give 3 decimal places.
Answer: 1.681
$\log 48 = \log(2^4 \cdot 3) = 4 \log 2 + \log 3 \approx 4 \cdot 0.301 + 0.477 = 1.204 + 0.477 = 1.681$ .
16.
Evaluate $3 \log_2 4 - \frac{1}{2} \log_2 16$ by condensing it into one logarithm.
Answer: 4
1. $3 \log_2 4 = \log_2 4^3 = \log_2 ?$ 64
2. $\frac{1}{2} \log_2 16 = \log_2 16^{1/2} = \log_2 ?$ 4
3. Quotient rule: $\log_2 64 - \log_2 4 = \log_2 ?$ 16
4. What is $\log_2 16$ ? 4
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.
Answer: 2.1385
1. $72 = 2^a \cdot 3^2$ . What is $a$ ? 3
2. $\ln 72 = 3 \ln 2 + 2 \ln 3$ . Compute $3 \cdot 0.693 + 2 \cdot 1.099$ . 4.277
3. $\ln \sqrt{72} = \frac{1}{2} \ln 72$ . Compute half of $4.277$ . 2.1385
18.
How many digits does $3^{50}$ have? Use $\log 3 \approx 0.477$ .
Answer: 24
1. Power rule: $\log(3^{50}) = 50 \log 3$ . Compute $50 \cdot 0.477$ . 23.85
2. What is the whole-number part of that result? 23
3. Add 1. How many digits does $3^{50}$ have? 24