Change of Base Formula

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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) $\dfrac{\log 5}{\log 20}$
2.
Use the change of base formula to evaluate $\log_2 10$ . Round to 2 decimal places.
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) $\log_2 7$
4.
Evaluate $\log_3 20$ with the log or ln key. Round to 2 decimal places.
5.
Evaluate $\log_6 100$ with common logarithms. Round to 2 decimal places.
6.
Evaluate $\log_2 30$ with natural logarithms. Round to 2 decimal places.
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) $3$
8.
Use base 2 to find the exact value of $\log_8 32$ . Write it as a fraction, like 7/4.
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.
10.
Solve $2^x = 50$ . The solution is $x = \log_2 50$ ; evaluate it with common logarithms. Round to 2 decimal places.
11.
Which is larger: $\log_2 12$ or $\log_3 30$ ?
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) $\dfrac{a}{b}$
13.
Find the exact value of $\log_{25} 125$ by changing to base 5. Give the answer as a decimal.
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) $\dfrac{\ln 9}{\ln 2}$
15.
Evaluate $\log_{1.5} 10$ . Round to 2 decimal places.
16.
An investment grows by 7 % per year (compounded yearly). After how many whole years has it at least doubled?
17.
Simplify $\log_2 3 \cdot \log_3 4 \cdot \log_4 8$ without a calculator.
18.
Evaluate $\log_4 60$ with natural logarithms. Round to 3 decimal places.