Quotient Rule for Logarithms

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Show your work for each problem.

1.
Which expression is equal to $\log_b\left(\dfrac{M}{N}\right)$ (for $M, N > 0$ )?
1. a) $\log_b M + \log_b N$
2. b) $\log_b M - \log_b N$
3. c) $\dfrac{\log_b M}{\log_b N}$
4. d) $\log_b N - \log_b M$
2.
Expand $\log_3\left(\dfrac{x}{7}\right)$ for $x > 0$ .
1. a) $\log_3 x - \log_3 7$
2. b) $\log_3 7 - \log_3 x$
3. c) $\dfrac{\log_3 x}{\log_3 7}$
4. d) $\log_3(x - 7)$
3.
Evaluate $\log_2 48 - \log_2 3$ .
4.
Evaluate $\log 5000 - \log 5$ (common logarithm, base 10).
5.
Evaluate $\log_3 162 - \log_3 2$ .
6.
Which statement is true?
1. a) $\dfrac{\log 100}{\log 10} = \log 10$
2. b) $\dfrac{\log 100}{\log 10} = 1$
3. c) $\log(100 - 10) = \log 100 - \log 10$
4. d) $\log 100 - \log 10 = \log 10$
7.
You know that $\log_2 x = 7$ . Find $\log_2\left(\dfrac{x}{32}\right)$ .
8.
Evaluate $\log_6 72 - \log_6 2$ .
9.
Write $\log 300 - \log 3$ as a single logarithm and evaluate it.
10.
Write $\log a + \log b - \log c$ as a single logarithm ( $a, b, c > 0$ ).
1. a) $\log(a + b - c)$
2. b) $\log(abc)$
3. c) $\log\left(\dfrac{ab}{c}\right)$
4. d) $\log\left(\dfrac{ac}{b}\right)$
11.
Use $\log 2 \approx 0.3$ and $\log 5 \approx 0.7$ (common logarithm, base 10) to approximate $\log\left(\dfrac{2}{5}\right)$ .
12.
Use $\log 3 \approx 0.477$ and $\log 12 \approx 1.079$ (common logarithm, base 10) to approximate $\log 4$ . Give 3 decimal places.
13.
Evaluate $\log_2 24 + \log_2 4 - \log_2 3$ .
14.
Given $\log_b x = 4$ , $\log_b y = 1.5$ and $\log_b z = 2$ , find $\log_b\left(\dfrac{xy}{z}\right)$ .
15.
Expand $\log_2\left(\dfrac{16}{xy}\right)$ completely ( $x, y > 0$ ).
1. a) $4 + \log_2 x - \log_2 y$
2. b) $4 - \log_2 x + \log_2 y$
3. c) $4 - \log_2 x - \log_2 y$
4. d) $\dfrac{4}{\log_2 x + \log_2 y}$
16.
Evaluate $\log_5 50 + \log_5 10 - \log_5 4$ .
17.
Given $\log_b M = 6$ , $\log_b N = 2.5$ and $\log_b P = 1$ , find $\log_b\left(\dfrac{M}{NP}\right)$ .
18.
Show that $\dfrac{\log 1000}{\log 10}$ is not the same as $\log\left(\dfrac{1000}{10}\right)$ : what is $\dfrac{\log 1000}{\log 10} - \log\left(\dfrac{1000}{10}\right)$ ?