Quotient Rule for Logarithms

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) $\log_b N - \log_b M$
Answer: $\log_b M - \log_b N$
Quotient rule: $\log_b\left(\dfrac{M}{N}\right) = \log_b M - \log_b N$ . The quotient of logarithms $\dfrac{\log_b M}{\log_b N}$ is a different thing, and $\log_b N - \log_b M$ has the wrong order.
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) $\log_3(x - 7)$
Answer: $\log_3 x - \log_3 7$
The numerator is $x$ and the denominator is $7$ , so $\log_3\left(\dfrac{x}{7}\right) = \log_3 x - \log_3 7$ .
3.
Evaluate $\log_2 48 - \log_2 3$ .
Answer: 4
$\log_2 48 - \log_2 3 = \log_2\left(\dfrac{48}{3}\right) = \log_2 16 = 4$ , because $2^4 = 16$ .
4.
Evaluate $\log 5000 - \log 5$ (common logarithm, base 10).
Answer: 3
$\log 5000 - \log 5 = \log\left(\dfrac{5000}{5}\right) = \log 1000 = 3$ , because $10^3 = 1000$ .
5.
Evaluate $\log_3 162 - \log_3 2$ .
Answer: 4
1. Condense: $\log_3 162 - \log_3 2 = \log_3 \square$ . What number goes in the box? 81
2. $3^{\square} = 81$ . What is the exponent, so what is $\log_3 81$ ? 4
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) $\log 100 - \log 10 = \log 10$
Answer: $\log 100 - \log 10 = \log 10$
$\log 100 - \log 10 = 2 - 1 = 1 = \log 10$ (quotient rule: $\log\left(\dfrac{100}{10}\right) = \log 10$ ). The others fail: $\dfrac{\log 100}{\log 10} = 2$ , and $\log 90 \approx 1.954 \neq 1$ .
7.
You know that $\log_2 x = 7$ . Find $\log_2\left(\dfrac{x}{32}\right)$ .
Answer: 2
1. Expand: $\log_2\left(\dfrac{x}{32}\right) = \log_2 x - \log_2 32$ . What is $\log_2 32$ ? 5
2. Substitute $\log_2 x = 7$ : what is $7 - 5$ ? 2
8.
Evaluate $\log_6 72 - \log_6 2$ .
Answer: 2
$\log_6 72 - \log_6 2 = \log_6\left(\dfrac{72}{2}\right) = \log_6 36 = 2$ ( $6^2 = 36$ ).
9.
Write $\log 300 - \log 3$ as a single logarithm and evaluate it.
Answer: 2
1. $\log 300 - \log 3 = \log \square$ . What number goes in the box? 100
2. What is $\log 100$ ? 2
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) $\log\left(\dfrac{ac}{b}\right)$
Answer: $\log\left(\dfrac{ab}{c}\right)$
Product rule: $\log a + \log b = \log(ab)$ . Quotient rule: $\log(ab) - \log c = \log\left(\dfrac{ab}{c}\right)$ .
11.
Use $\log 2 \approx 0.3$ and $\log 5 \approx 0.7$ (common logarithm, base 10) to approximate $\log\left(\dfrac{2}{5}\right)$ .
Answer: -0.4
1. $\log\left(\dfrac{2}{5}\right) = \log 2 - \log 5$ . What is the value of the first term, $\log 2$ ? 0.3
2. Subtract: what is $0.3 - 0.7$ ? -0.4
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.
Answer: 0.602
$4 = \dfrac{12}{3}$ , so $\log 4 = \log 12 - \log 3 \approx 1.079 - 0.477 = 0.602$ (a calculator gives $0.60206$ ).
13.
Evaluate $\log_2 24 + \log_2 4 - \log_2 3$ .
Answer: 5
1. Product rule: $\log_2 24 + \log_2 4 = \log_2 \square$ . What number goes in the box? 96
2. Quotient rule: $\log_2 96 - \log_2 3 = \log_2 \square$ . What number goes in the box? 32
3. What is $\log_2 32$ ? 5
14.
Given $\log_b x = 4$ , $\log_b y = 1.5$ and $\log_b z = 2$ , find $\log_b\left(\dfrac{xy}{z}\right)$ .
Answer: 3.5
1. Numerator: $\log_b(xy) = \log_b x + \log_b y$ . What is its value? 5.5
2. Subtract $\log_b z$ : what is $5.5 - 2$ ? 3.5
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) $\dfrac{4}{\log_2 x + \log_2 y}$
Answer: $4 - \log_2 x - \log_2 y$
$\log_2\left(\dfrac{16}{xy}\right) = \log_2 16 - (\log_2 x + \log_2 y) = 4 - \log_2 x - \log_2 y$
16.
Evaluate $\log_5 50 + \log_5 10 - \log_5 4$ .
Answer: 3
$\log_5 50 + \log_5 10 - \log_5 4 = \log_5\left(\dfrac{500}{4}\right) = \log_5 125 = 3$ ( $5^3 = 125$ )
17.
Given $\log_b M = 6$ , $\log_b N = 2.5$ and $\log_b P = 1$ , find $\log_b\left(\dfrac{M}{NP}\right)$ .
Answer: 2.5
1. Denominator: $\log_b(NP) = \log_b N + \log_b P$ . What is its value? 3.5
2. $\log_b\left(\dfrac{M}{NP}\right) = \log_b M - \log_b(NP)$ . What is $6 - 3.5$ ? 2.5
18.
Show that $\dfrac{\log 1000}{\log 10}$ is not the same as $\log\left(\dfrac{1000}{10}\right)$ : what is $\dfrac{\log 1000}{\log 10} - \log\left(\dfrac{1000}{10}\right)$ ?
Answer: 1
1. What is $\dfrac{\log 1000}{\log 10}$ ? 3
2. What is $\log\left(\dfrac{1000}{10}\right)$ ? 2
3. What is the difference $3 - 2$ ? 1