What is a logarithm?
A logarithm answers the question "to what power must the base be raised?" If ax = n then loga n = x. For example log2 8 = 3 because 23 = 8. The base a must be positive and not equal to 1.
Key formula
loga(mn) = loga m + loga n
loga(m/n) = loga m − loga n
loga(mk) = k loga m
loga a = 1, loga 1 = 0
Using the laws
The product law turns multiplication into addition, the quotient law turns division into subtraction, and the power law brings an index to the front:
- log2 4 + log2 8 = log2(4 × 8) = log2 32 = 5
- log3 27 − log3 3 = log3 9 = 2
- 2 log2 3 = log2 32 = log2 9
Worked example
Given log2 5 = 2.32, find log2 10.
Write 10 = 2 × 5, then apply the product law: log2 10 = log2 2 + log2 5 = 1 + 2.32 = 3.32.
Solving log equations
Convert between log and index form. To solve log2 x = 4, rewrite as x = 24 = 16.
Remember
- loga a = 1 and loga 1 = 0.
- The laws only apply to logs of the same base.
- You cannot take the log of zero or a negative number.