Product Rule for Logarithms

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Show your work for each problem.

1.
Which expression is equal to $\log_b(MN)$ (with $M, N > 0$ )?
1. a) $\log_b M + \log_b N$
2. b) $\log_b M \cdot \log_b N$
3. c) $\log_b(M + N)$
4. d) $\log_b M - \log_b N$
2.
Use the product rule: $\log_2(4 \cdot 8) = \log_2 4 + \log_2 8 = \;?$
3.
Expand $\log_3(5x)$ for $x > 0$ .
1. a) $\log_3(5 + x)$
2. b) $5 \log_3 x$
3. c) $\log_3 5 + \log_3 x$
4. d) $\log_3 5 \cdot \log_3 x$
4.
Calculate $\log 2 + \log 50$ (common logarithm, base 10) without a calculator.
5.
Write $\log_6 4 + \log_6 9$ as a single logarithm and evaluate it.
6.
Check that $\log_2 32 = \log_2 4 + \log_2 8$ .
7.
Which statement is true?
1. a) $\log(10 + 10) = 2$
2. b) $\log(2 \cdot 5) = \log 2 + \log 5$
3. c) $\log(6 + 4) = \log 6 + \log 4$
4. d) $\log 2 \cdot \log 5 = \log 10$
8.
Use $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$ to approximate $\log 6$ . Give the answer with 3 decimal places.
9.
If $\log_b M = 3.2$ and $\log_b N = 1.5$ , what is $\log_b(MN)$ ?
10.
Approximate $\log 12$ using $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$ . Give the answer with 3 decimal places.
11.
Expand $\log_2(8xy) = \log_2 8 + \log_2 x + \log_2 y$ and evaluate it for $x = 4$ and $y = 16$ .
12.
Condense $\log_4 2 + \log_4 32$ into one logarithm and evaluate it.
13.
Condense $\log_5 x + \log_5(x + 3)$ into a single logarithm.
1. a) $2\log_5 x + 3$
2. b) $\log_5(x^2 + 3x)$
3. c) $\log_5(x^2 + 3)$
4. d) $\log_5(2x + 3)$
14.
For which values of $x$ is $\log x + \log(x - 4) = \log(x^2 - 4x)$ true?
1. a)For $x > 4$
2. b)For all real numbers $x$
3. c)For $x > 0$
4. d)For $x < 0$ or $x > 4$
15.
Use $\log 2 \approx 0.301$ to approximate $\log 2000$ . Give the answer with 3 decimal places.
16.
Condense $\log_3 x + \log_3 9 + \log_3 x$ into one logarithm $\log_3(9x^2)$ and evaluate it for $x = 3$ .
17.
A sound has a level of $50$ decibels: $L = 10 \log\left(\frac{I}{I_0}\right) = 50$ . Its intensity is multiplied by $100$ . Use $10 \log\left(100 \cdot \frac{I}{I_0}\right) = 10 \log 100 + 10 \log\frac{I}{I_0}$ to find the new level.
18.
Use $\log 2 \approx 0.301$ , $\log 3 \approx 0.477$ and $\log 7 \approx 0.845$ to approximate $\log 42$ . Give the answer with 3 decimal places.