Properties of Logarithms

Show your work for each problem.

1.
What is $\log_6 1$ ?
1. a)it is undefined
2. b)0
3. c)6
4. d)1
Answer: 0
$6^0 = 1$ , so $\log_6 1 = 0$ . The log of $1$ is $0$ for every allowed base.
2.
Evaluate $\log_8 8$ .
Answer: 1
$8^1 = 8$ , so $\log_8 8 = 1$ . The log of the base is always $1$ .
3.
Evaluate $\log_5 1 + \log_5 5 + \log_5 25$ .
Answer: 3
1. What is $\log_5 1$ ? 0
2. What is $\log_5 5$ ? 1
3. Write $25$ as a power of $5$ . What is $\log_5 25$ ? 2
4. Add the three values. 3
4.
Simplify $\log_3(3^7)$ .
Answer: 7
The base of the logarithm and the base of the power are both $3$ , so they cancel and the exponent remains: $\log_3(3^7) = 7$ .
5.
Simplify $5^{\log_5 9}$ .
1. a)9
2. b)45
3. c)5
4. d) $\log_5 9$
Answer: 9
$\log_5 9$ is the exponent that turns $5$ into $9$ . Raising $5$ to that exponent gives $9$ .
6.
Evaluate $\log 10000$ (common logarithm, base 10).
Answer: 4
1. Write $10000$ as a power of $10$ : $10000 = 10^{?}$ 4
2. Use $\log(10^x) = x$ : what is $\log 10000$ ? 4
7.
Evaluate $\log_2 32$ .
Answer: 5
$32 = 2^5$ , so $\log_2 32 = \log_2(2^5) = 5$ .
8.
Which of these logarithms is undefined?
1. a) $\log_5 5$
2. b) $\log_3(-9)$
3. c) $\log_4 \frac{1}{4}$
4. d) $\log_2 1$
Answer: $\log_3(-9)$
The argument $-9$ is negative, and no power of $3$ is negative, so $\log_3(-9)$ is undefined. The others are $0$ , $-1$ and $1$ .
9.
For which $x$ is $\log_3(x - 4)$ defined? Find the smallest whole number $x$ that is allowed.
Answer: 5
1. The argument $x - 4$ must be greater than which number? 0
2. Solve $x - 4 > 0$ : $x$ must be greater than which number? 4
3. What is the smallest whole number $x$ that is allowed? 5
10.
Evaluate $\log_5 \frac{1}{25}$ .
Answer: -2
$\frac{1}{25} = 5^{-2}$ , so $\log_5 \frac{1}{25} = \log_5(5^{-2}) = -2$ . The argument is positive, the value is negative.
11.
Simplify $2^{\log_2 5} \cdot 3^{\log_3 4}$ .
Answer: 20
1. What is $2^{\log_2 5}$ ? 5
2. What is $3^{\log_3 4}$ ? 4
3. Multiply the two results. 20
12.
If $\log_4 M = \log_4 20$ , what is $M$ ?
1. a)4
2. b)5
3. c)20
4. d)80
Answer: 20
With the same base, equal logarithms mean equal arguments, so $M = 20$ .
13.
Solve $\log_3(x + 4) = \log_3 10$ .
Answer: 6
1. Set the arguments equal: $x + 4 = ?$ 10
2. Solve for $x$ . 6
3. Check: what is the argument $x + 4$ for this $x$ ? 10
14.
Evaluate $\log_3 \frac{1}{9} + 6^{\log_6 11}$ .
Answer: 9
1. Write $\frac{1}{9}$ as a power of $3$ : $\frac{1}{9} = 3^{?}$ . So what is $\log_3 \frac{1}{9}$ ? -2
2. What is $6^{\log_6 11}$ ? 11
3. Add the two values. 9
15.
Which statement is true for every allowed base $b$ ?
1. a) $\log_b 0 = 0$
2. b) $\log_b(b^x) = x$ for every real $x$
3. c) $\log_b 1 = 1$
4. d) $b^{\log_b x} = x$ for every real $x$
Answer: $\log_b(b^x) = x$ for every real $x$
$\log_b(b^x) = x$ holds for every real $x$ , because $b^x$ is always positive. $\log_b 1 = 0$ , not $1$ ; $b^{\log_b x} = x$ needs $x > 0$ ; and $\log_b 0$ is undefined.
16.
Evaluate $\log_2(8^4)$ . First write $8$ as a power of $2$ .
Answer: 12
$8^4 = (2^3)^4 = 2^{12}$ , so $\log_2(8^4) = \log_2(2^{12}) = 12$ .
17.
Solve $\log_5(x^2 - 6) = \log_5 x$ .
Answer: 3
1. Setting the arguments equal gives $x^2 - x - 6 = 0$ . What is the positive root? 3
2. What is the negative root? -2
3. How many of the two candidates make both arguments positive? 1
4. What is the solution of the equation? 3
18.
Evaluate $\log_4 8$ by writing $8$ as a power of $4$ .
Answer: 3/2
1. Write $2$ as a power of $4$ : $2 = 4^{?}$ (as a fraction, like 1/3) 1/2
2. Now $8 = 2^3 = 4^{?}$ (as a fraction) 3/2
3. Use $\log_b(b^x) = x$ : what is $\log_4 8$ ? 3/2