About Logarithm
A logarithm is just an index written the other way round. Asking for the logarithm of a number to a given base is the same as asking what power the base must be raised to. Once that is clear, the rest of the topic is three laws and a table.
What you need to understand
- The logarithm of a number to a base is the power to which the base must be raised to give that number.
- The logarithm of the base itself is always 1, and the logarithm of 1 is always 0, whatever the base.
- Adding two logarithms with the same base multiplies the numbers inside them.
- Subtracting two logarithms divides the numbers inside them, and multiplying a logarithm by a number raises the number inside to that power.
- A logarithm is only defined for positive numbers.
- Values that are not exact powers of ten are found from a four figure table, which is why exam questions supply the value of log 2 or log 3.
Formulas to remember
How to work through these questions
- Rewrite every logarithm question as a power statement before trying to calculate anything.
- For a number that is a power of the base, the answer is the index of that power.
- When the question supplies a table value, express the number asked about as a power of the number given.
- To find an unknown base, turn the statement into a power equation and take the matching root.
- Check the size of the answer, because the logarithm of a number larger than the base must be greater than 1.
Mistakes that cost marks
- Multiplying the logarithms when the numbers should be multiplied, instead of adding them.
- Taking the logarithm of the base rather than of the number asked about.
- Forgetting that the logarithm of 1 is zero for every base.
- Trying to take the logarithm of a negative number, which has no value.
- Reading the table value as the answer when the question asks about a higher power.
Worked example
If log 2 = 0.3010, find log 8.
- Write 8 as a power of 2: 8 = 2 cubed.
- So log 8 = 3 x log 2.
- That is 3 x 0.3010 = 0.9030.
- Check: 8 is larger than 2, so its logarithm must be larger than 0.3010, and 0.9030 is.
Answer: 0.9030
Practice questions with answers
A few Logarithm questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
What is log of 4096 to the base 4?
Answer: Option B — with explanation
A logarithm answers the question: to what power must the base be raised?
$4^{6} = 4096$, so the logarithm of 4096 to the base 4 is 6.
Common mistakes
- divided the number by the base instead of taking the logarithm: 1024
- counted one power too many: 7
- counted one power too few: 5
Question 2
The value of log 13 to the base 10 is 1.1139. Find log 2197 to the base 10.
-
A
3.3417
-
B
1.1139
-
C
1.6708
-
D
4.4556
Answer: Option A — with explanation
$2197 = 13^{3}$, so its logarithm is 3 times the logarithm of 13.
$\log 2197 = 3 \times 1.1139 = 3.3417$.
Common mistakes
- gave the logarithm of the base instead of of the power: 1.1139
- halved the result instead of taking the full multiple: 1.6708
- used one power too many: 4.4556
Question 3
The logarithm of 36 to the base x is 2. Find x.
Answer: Option A — with explanation
The statement means $x^{2} = 36$.
So x is the 2th root of 36, and since $6^{2} = 36$, $x = 6$.
Common mistakes
- gave the number itself as the base: 36
- divided the number by the logarithm: 18
- gave the logarithm as the base: 2
Question 4
Find the value of the logarithm of 4 to the base 2.
Answer: Option D — with explanation
A logarithm answers the question: to what power must the base be raised?
$2^{2} = 4$, so the logarithm of 4 to the base 2 is 2.
Common mistakes
- counted one power too few: 1
- counted one power too many: 3
- reported three times the correct value: 6
Question 5
The value of log 19 to the base 10 is 1.2788. Find log 361 to the base 10.
-
A
2.5576
-
B
3.2788
-
C
3.8364
-
D
1.2788
Answer: Option A — with explanation
$361 = 19^{2}$, so its logarithm is 2 times the logarithm of 19.
$\log 361 = 2 \times 1.2788 = 2.5576$.
Common mistakes
- added the index to the logarithm instead of multiplying: 3.2788
- gave the logarithm of the base instead of of the power: 1.2788
- used one power too many: 3.8364
Frequently asked questions
What is a logarithm in plain words?
It is the answer to the question, what power must the base be raised to. The logarithm of 1000 to the base 10 is 3, because 10 cubed is 1000.
Why is the logarithm of 1 always zero?
Because any number raised to the power zero is 1. Whatever the base, the power needed to reach 1 is zero.
What is the difference between log and ln?
log usually means base 10, which is what examination questions use, while ln means the natural logarithm with base e.
Can I take the logarithm of a negative number?
No. Powers of a positive base are always positive, so no power of the base gives a negative number.
