Product Rule for Logarithms

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) $\log_b M - \log_b N$
Answer: $\log_b M + \log_b N$
By the product rule, $\log_b(MN) = \log_b M + \log_b N$ . The log of a product is the sum of the logs, not their product.
2.
Use the product rule: $\log_2(4 \cdot 8) = \log_2 4 + \log_2 8 = \;?$
Answer: 5
$\log_2 4 = 2$ and $\log_2 8 = 3$ , so the sum is $5$ . Check: $4 \cdot 8 = 32 = 2^5$ .
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) $\log_3 5 \cdot \log_3 x$
Answer: $\log_3 5 + \log_3 x$
$\log_3(5x) = \log_3 5 + \log_3 x$ . The factor $5$ does not move in front of the log (that would need a power, $x^5$ ).
4.
Calculate $\log 2 + \log 50$ (common logarithm, base 10) without a calculator.
Answer: 2
$\log 2 + \log 50 = \log 100 = 2$ , because $10^2 = 100$ .
5.
Write $\log_6 4 + \log_6 9$ as a single logarithm and evaluate it.
Answer: 2
1. Multiply the arguments: $4 \cdot 9 = \;?$ 36
2. What is $\log_6 36$ ? 2
6.
Check that $\log_2 32 = \log_2 4 + \log_2 8$ .
Answer: 5
1. What is $\log_2 4$ ? 2
2. What is $\log_2 8$ ? 3
3. What is $\log_2 32$ ? (It should equal the sum of the two results.) 5
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) $\log 2 \cdot \log 5 = \log 10$
Answer: $\log(2 \cdot 5) = \log 2 + \log 5$
$\log(2 \cdot 5) = \log 2 + \log 5$ is the product rule ( $= \log 10 = 1$ ). The others are false: $\log(6+4) = 1$ but $\log 6 + \log 4 \approx 1.380$ ; $\log 2 \cdot \log 5 \approx 0.210$ ; $\log 20 \approx 1.301$ , not $2$ .
8.
Use $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$ to approximate $\log 6$ . Give the answer with 3 decimal places.
Answer: 0.778
$\log 6 = \log 2 + \log 3 \approx 0.301 + 0.477 = 0.778$ . (A calculator gives $0.77815\ldots$ )
9.
If $\log_b M = 3.2$ and $\log_b N = 1.5$ , what is $\log_b(MN)$ ?
Answer: 4.7
$\log_b(MN) = \log_b M + \log_b N = 3.2 + 1.5 = 4.7$ . You do not need to know $b$ , $M$ or $N$ .
10.
Approximate $\log 12$ using $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$ . Give the answer with 3 decimal places.
Answer: 1.079
1. $12 = 2 \cdot 2 \cdot 3$ . How many factors $2$ are there? 2
2. What is $\log 2 + \log 2$ ? 0.602
3. Add $\log 3$ : what is $\log 12$ ? 1.079
11.
Expand $\log_2(8xy) = \log_2 8 + \log_2 x + \log_2 y$ and evaluate it for $x = 4$ and $y = 16$ .
Answer: 9
1. What is $\log_2 8$ ? 3
2. What is $\log_2 x$ for $x = 4$ ? 2
3. What is $\log_2 y$ for $y = 16$ ? 4
4. Add the three results. What is $\log_2(8xy)$ ? 9
12.
Condense $\log_4 2 + \log_4 32$ into one logarithm and evaluate it.
Answer: 3
1. Multiply the arguments: $2 \cdot 32 = \;?$ 64
2. What is $\log_4 64$ ? 3
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) $\log_5(2x + 3)$
Answer: $\log_5(x^2 + 3x)$
$\log_5 x + \log_5(x+3) = \log_5\big(x(x+3)\big) = \log_5(x^2 + 3x)$ , valid for $x > 0$ .
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 $x > 4$
Answer: For $x > 4$
The left side exists only when $x > 0$ and $x > 4$ , so for $x > 4$ . The right side also exists for $x < 0$ (for example $x = -1$ gives $\log 5$ ), but then $\log(-1)$ on the left does not exist. The equality holds exactly for $x > 4$ .
15.
Use $\log 2 \approx 0.301$ to approximate $\log 2000$ . Give the answer with 3 decimal places.
Answer: 3.301
$\log 2000 = \log 2 + \log 1000 \approx 0.301 + 3 = 3.301$ . Multiplying by $1000$ just adds $3$ to the log.
16.
Condense $\log_3 x + \log_3 9 + \log_3 x$ into one logarithm $\log_3(9x^2)$ and evaluate it for $x = 3$ .
Answer: 4
1. What is the argument $9x^2$ for $x = 3$ ? 81
2. What is $\log_3 81$ ? 4
3. Check with the original sum: $\log_3 3 + \log_3 9 + \log_3 3 = \;?$ 4
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.
Answer: 70
1. What is $\log 100$ ? 2
2. What is $10 \log 100$ ? 20
3. What is the new level in decibels? 70
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.
Answer: 1.623
1. $42 = 2 \cdot 3 \cdot \;?$ 7
2. What is $\log 2 + \log 3$ ? 0.778
3. Add $\log 7$ : what is $\log 42$ ? 1.623