Definition of a Logarithm

Show your work for each problem.

1.
What does mean in exponential form?
1. a)
2. b)
3. c)
4. d)
Answer:
means 2 raised to what power equals 8. The answer is 3, so .
2.
Evaluate .
Answer: 2
because .
3.
In the expression , what is the base?
1. a)125
2. b)5
3. c)3
4. d)25
Answer: 5
In , the base is 5 (the subscript), and 125 is the argument (inside parentheses).
4.
Evaluate .
Answer: 2
because .
5.
Write in logarithmic form.
Answer: log_4(64) = 3
1. What is the base in ? 4
2. What is the exponent? 3
3. What is the result (argument)? 64
6.
Evaluate (common logarithm).
Answer: 2
because .
7.
What is for any valid base ?
1. a)
2. b)0
3. c)1
4. d)undefined
Answer: 0
because for any positive base .
8.
Evaluate .
Answer: 6
1. What is the question we are asking? 2 to what power equals 64
2. List powers of 2: 64
3. So 6
9.
Evaluate .
Answer: 3
because .
10.
What is for any valid base ?
1. a)1
2. b)0
3. c)
4. d)
Answer: 0
because . Any number raised to the first power is itself.
11.
Write in exponential form.
Answer: 7^2 = 49
1. What is the base of the logarithm? 7
2. What is the argument? 49
3. What is the value (exponent)? 2
12.
Evaluate .
Answer: 4
because . Notice: 10000 has 4 zeros!
13.
Which of the following is NOT a valid logarithm?
1. a)
2. b)
3. c)
4. d)
Answer:
is not valid because we cannot take the logarithm of a negative number.
14.
If , find the value of .
Answer: 3
1. Convert to exponential form: 4
2. We need . What is as a power of some number? 3^4
3. Since , what is ? 3
15.
Evaluate .
Answer: 7
1. Start listing powers of 2: , , 8
2. Continue: , , 64
3. Finally: 128
4. So 7
16.
Why is undefined?
1. a)5 is not a perfect power
2. b)Logarithms only work with base 10
3. c) for all , never 5
4. d)The base must be greater than 5
Answer: for all , never 5
raised to any power always equals . There is no exponent that makes , so is undefined.