Introduction to Logarithms

Show your work for each problem.

1.
What does equal?
1. a)4
2. b)3
3. c)2
4. d)8
Answer: 3
because . The logarithm tells us the exponent needed.
2.
Which expression is equivalent to ?
1. a)
2. b)
3. c)
4. d)
Answer:
converts to . The base (10) raised to the exponent (2) equals the argument (100).
3.
Evaluate . (Remember: means )
Answer: 1
because . The logarithm of any base to itself is always 1.
4.
What is ?
Answer: 0
because . The log of 1 is always 0 for any valid base.
5.
What is ?
1. a)
2. b)0
3. c)2.718
4. d)1
Answer: 1
because . The natural log of is 1.
6.
Evaluate .
Answer: 3
because .
7.
Evaluate .
Answer: 4
because . For powers of 10, the log equals the number of zeros.
8.
Write in logarithmic form.
1. a)
2. b)
3. c)
4. d)
Answer:
becomes . The base (5) is the subscript, the result (125) is the argument, and the exponent (3) is the answer.
9.
Convert to exponential form.
Answer: 4^3 = 64
1. What is the base of this logarithm? 4
2. What is the argument (inside the log)? 64
3. What is the value of the logarithm? 3
4. Write in form: base^exponent = argument 4^3 = 64
10.
Evaluate step by step.
Answer: 5
1. What question does ask? 2 to what power equals 32
2. Calculate: ? 2
3. Calculate: ? 4
4. Calculate: ? 8
5. Calculate: ? 16
6. Calculate: ? 32
7. So ? 5
11.
Which statement is TRUE about logarithms?
1. a) for any base
2. b)
3. c) for any valid base
4. d)
Answer: for any valid base
is true because for any valid base. The other options are false: is undefined, logs don't distribute over addition, and , not .
12.
Evaluate .
Answer: 7
because is , and when the exponent is 7. The natural log and the exponential with base are inverse functions.
13.
Evaluate by expressing 256 as a power of 4.
Answer: 4
1. What is ? 4
2. What is ? 16
3. What is ? 64
4. What is ? 256
5. So ? 4
14.
A sound is measured at 80 dB. Another sound is measured at 50 dB. How many times more intense is the first sound? Use .
Answer: 1000
1. What is the difference in decibels? 30
2. Divide by 10 to get the exponent 3
3. Calculate 1000
15.
Which of the following is undefined?
1. a)
2. b)
3. c)
4. d)
Answer:
is undefined because the argument must be positive. The others are defined: , , .
16.
Convert to logarithmic form and verify.
Answer: log_2(1024) = 10
1. In , what is the base? 2
2. What is the exponent? 10
3. What is the result? 1024
4. Write in log form: log_2(1024) = 10
17.
If , what is ?
Answer: 3
means . Since , we have .