Logarithms — Challenge
A logarithm is the exponent — nothing more, and nothing less.
What these problems cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Integer — A whole number, positive, negative, or zero.
- Base — The number being multiplied. In 53 the base is 5.
- Exponent — How many times the base is used as a factor. In 53 the exponent is 3.
- Zeros — The x values where y = 0.
- Logarithm — logb y is the exponent that turns b into y.
- Quotient — The answer to a division.
Problem 1
What is log28?
- 2
- 3
- 4
- 8
- 16
Worked answer
3
A logarithm asks: to what power must the base be raised to give the number?
23 = 8, so log28 = 3.
Problem 2
What is log10100?
- 1
- 2
- 10
- 100
- 0
Worked answer
2
Ask what power of 10 gives 100.
102 = 100, so the logarithm is 2.
Problem 3
What is log51?
- 0
- 1
- 5
- -1
- It is undefined
Worked answer
0
Ask what power of 5 gives 1.
Any non-zero number to the power 0 is 1, so the answer is 0.
This holds for every base: logb1 = 0.
Problem 4
What is log327?
- 2
- 3
- 9
- 27
- 1
Worked answer
3
Ask what power of 3 gives 27.
33 = 27, so the logarithm is 3.
Here the base and the answer happen to coincide, which is a coincidence of this example.
Problem 5
What is log102+log105?
- 0
- 1
- 7
- 10
- log107
Worked answer
1
Adding logarithms of the same base corresponds to multiplying the numbers: log 2+log 5 = log(2· 5).
2· 5 = 10, and log1010 = 1.
Adding the numbers to get log 7 is the mistake the options invite.
Problem 6
If 2x=16, what is x?
- 2
- 3
- 4
- 8
- 16
Worked answer
4
List the powers of 2: 2, 4, 8, 16, reached at exponents 1, 2, 3, 4.
So 24 = 16 and x = 4.
In logarithm notation this says log216 = 4.
Problem 7
What is log218?
- -3
- 3
- -8
- 8
- 13
Worked answer
-3
Ask what power of 2 gives 18.
A reciprocal needs a negative exponent: 2-3 = 123 = 18.
So the logarithm is -3. Logarithms of numbers below 1 are negative.
Problem 8
What is logbb for any valid base b?
- 0
- 1
- b
- It is undefined
- -1
Worked answer
1
Ask what power of b gives b itself.
Any number to the power 1 is itself, so the answer is 1.
Together with logb1 = 0, this is one of the two logarithm facts worth knowing by heart.
Problem 9
What is log416?
- 2
- 4
- 8
- 16
- 0.5
Worked answer
2
Ask what power of 4 gives 16.
42 = 16, so the logarithm is 2.
Note that log216 = 4: changing the base changes the answer.