Change of Base Formula

Show your work for each problem.

1.
Which expression is equal to $\log_5 20$ ?
1. a) $\log 20 \cdot \log 5$
2. b) $\log \dfrac{20}{5}$
3. c) $\dfrac{\log 20}{\log 5}$
4. d) $\dfrac{\log 5}{\log 20}$
Answer: $\dfrac{\log 20}{\log 5}$
By the change of base formula, $\log_5 20 = \dfrac{\log 20}{\log 5} \approx 1.86$ . The upside down fraction is wrong, and $\log \dfrac{20}{5} = \log 20 - \log 5 = \log 4$ is a different number.
2.
Use the change of base formula to evaluate $\log_2 10$ . Round to 2 decimal places.
Answer: 3.32
$\log_2 10 = \dfrac{\log 10}{\log 2} = \dfrac{1}{0.30103} \approx 3.32193 \approx 3.32$ . Check: $2^3 = 8$ and $2^4 = 16$ , so the answer lies between 3 and 4.
3.
Which logarithm is equal to $\dfrac{\ln 7}{\ln 2}$ ?
1. a) $\ln 5$
2. b) $\log_7 2$
3. c) $\log_2 14$
4. d) $\log_2 7$
Answer: $\log_2 7$
Reading the formula backwards: $\dfrac{\ln 7}{\ln 2} = \log_2 7 \approx 2.81$ . The base 2 is in the denominator and the argument 7 on top. $\log_7 2$ is the upside down version, and $\ln 5$ comes from wrongly subtracting $7 - 2$ .
4.
Evaluate $\log_3 20$ with the log or ln key. Round to 2 decimal places.
Answer: 2.73
$\log_3 20 = \dfrac{\log 20}{\log 3} \approx \dfrac{1.30103}{0.47712} \approx 2.72683 \approx 2.73$ . With ln: $\dfrac{2.99573}{1.09861}$ gives the same result.
5.
Evaluate $\log_6 100$ with common logarithms. Round to 2 decimal places.
Answer: 2.57
1. What is $\log 100$ ? 2
2. What is $\log 6$ ? Round to 4 decimal places. 0.7782
3. Divide $\log 100$ by $\log 6$ . Round to 2 decimal places. 2.57
6.
Evaluate $\log_2 30$ with natural logarithms. Round to 2 decimal places.
Answer: 4.91
1. What is $\ln 30$ ? Round to 4 decimal places. 3.4012
2. What is $\ln 2$ ? Round to 4 decimal places. 0.6931
3. Divide $\ln 30$ by $\ln 2$ . Round to 2 decimal places. 4.91
7.
What is the exact value of $\log_9 27$ ? (Hint: choose base 3.)
1. a) $\dfrac{3}{2}$
2. b) $\dfrac{1}{3}$
3. c) $\dfrac{2}{3}$
4. d) $3$
Answer: $\dfrac{3}{2}$
$\log_9 27 = \dfrac{\log_3 27}{\log_3 9} = \dfrac{3}{2}$ . Check: $9^{3/2} = (\sqrt{9})^3 = 3^3 = 27$ .
8.
Use base 2 to find the exact value of $\log_8 32$ . Write it as a fraction, like 7/4.
Answer: 5/3
$\log_8 32 = \dfrac{\log_2 32}{\log_2 8} = \dfrac{5}{3}$ . Check: $8^{5/3} = (\sqrt[3]{8})^5 = 2^5 = 32$ .
9.
You are told that $\log_3 7 \approx 1.7712$ . Use the reciprocal rule to find $\log_7 3$ . Round to 4 decimal places.
Answer: 0.5646
Swapping base and argument gives the reciprocal: $\log_7 3 = \dfrac{1}{1.7712} \approx 0.56459 \approx 0.5646$ . Makes sense: $3 < 7$ , so $\log_7 3$ is below 1.
10.
Solve $2^x = 50$ . The solution is $x = \log_2 50$ ; evaluate it with common logarithms. Round to 2 decimal places.
Answer: 5.64
1. What is $\log 50$ ? Round to 4 decimal places. 1.6990
2. What is $\log 2$ ? Round to 4 decimal places. 0.3010
3. Divide to find $x$ . Round to 2 decimal places. 5.64
11.
Which is larger: $\log_2 12$ or $\log_3 30$ ?
Answer: $\log_2 12$
1. Evaluate $\log_2 12$ . Round to 2 decimal places. 3.58
2. Evaluate $\log_3 30$ . Round to 2 decimal places. 3.10
3. Which logarithm is larger? $\log_2 12$
12.
For positive numbers $a \neq 1$ and $b \neq 1$ , what is $\log_b a \cdot \log_a b$ ?
1. a) $1$
2. b) $\log(ab)$
3. c) $0$
4. d) $\dfrac{a}{b}$
Answer: $1$
$\log_b a \cdot \log_a b = \dfrac{\log a}{\log b} \cdot \dfrac{\log b}{\log a} = 1$ . The two logarithms are reciprocals, for example $\log_2 8 \cdot \log_8 2 = 3 \cdot \dfrac{1}{3} = 1$ .
13.
Find the exact value of $\log_{25} 125$ by changing to base 5. Give the answer as a decimal.
Answer: 1.5
1. What is $\log_5 125$ ? 3
2. What is $\log_5 25$ ? 2
3. Divide the first result by the second. What is $\log_{25} 125$ as a decimal? 1.5
14.
Which expression is NOT equal to $\log_2 9$ ?
1. a) $\dfrac{\log 2}{\log 9}$
2. b) $\dfrac{1}{\log_9 2}$
3. c) $\dfrac{\log 9}{\log 2}$
4. d) $\dfrac{\ln 9}{\ln 2}$
Answer: $\dfrac{\log 2}{\log 9}$
$\dfrac{\log 9}{\log 2}$ and $\dfrac{\ln 9}{\ln 2}$ are the change of base formula ( $\approx 3.17$ ), and $\dfrac{1}{\log_9 2}$ is the reciprocal rule. $\dfrac{\log 2}{\log 9} = \log_9 2 \approx 0.32$ is the upside down fraction.
15.
Evaluate $\log_{1.5} 10$ . Round to 2 decimal places.
Answer: 5.68
$\log_{1.5} 10 = \dfrac{1}{\log 1.5} \approx \dfrac{1}{0.17609} \approx 5.67887 \approx 5.68$ . A base close to 1 grows slowly, so it takes a large exponent to reach 10: $1.5^{5.68} \approx 10$ .
16.
An investment grows by 7 % per year (compounded yearly). After how many whole years has it at least doubled?
Answer: 11
1. What is the yearly growth factor (the base of the logarithm)? 1.07
2. Evaluate $\dfrac{\log 2}{\log 1.07}$ . Round to 2 decimal places. 10.24
3. Interest is added once a year. After how many whole years has the investment at least doubled? 11
17.
Simplify $\log_2 3 \cdot \log_3 4 \cdot \log_4 8$ without a calculator.
Answer: 3
1. After cancelling, the numerator is $\ln$ of which number? 8
2. And the denominator is $\ln$ of which number? 2
3. What is the value of the whole product? 3
18.
Evaluate $\log_4 60$ with natural logarithms. Round to 3 decimal places.
Answer: 2.953
1. What is $\ln 60$ ? Round to 4 decimal places. 4.0943
2. What is $\ln 4$ ? Round to 4 decimal places. 1.3863
3. Divide $\ln 60$ by $\ln 4$ . Round to 3 decimal places. 2.953