A logarithm answers one question: what power do I raise this base to, in order to get this number? Fill in two of the three and this finds the third, showing the change of base rule where it applies.
A logarithm is the inverse of raising to a power. If 2⁶ = 64, then log₂(64) = 6. The log is asking: how many times do I multiply 2 by itself to reach 64? The answer is the exponent.
That is the whole idea. Every log question is an index question written the other way round, and switching between the two forms is usually the fastest way to see what a question wants.
These three come directly from the index laws, which is worth seeing rather than memorising blindly. Multiplying powers adds exponents, so multiplying inside a log adds the logs.
| Law | Rule | Comes from |
|---|---|---|
| Product | log(xy) = log x + log y | bᵐ × bⁿ = bᵐ⁺ⁿ |
| Quotient | log(x ÷ y) = log x − log y | bᵐ ÷ bⁿ = bᵐ⁻ⁿ |
| Power | log(xⁿ) = n log x | (bᵐ)ⁿ = bᵐⁿ |
| Change of base | log_b x = log x ÷ log b | lets you use any calculator |
Two useful facts fall out of these. log_b(1) = 0 for any base, because anything to the power zero is 1. And log_b(b) = 1, because a base to the power one is itself.
log(x + y) is not log x + log y. The product law applies to multiplication inside the log, not addition. There is no law that simplifies the log of a sum, and trying to split one is probably the single most common error in this topic.
Similarly, log(x) ÷ log(y) is not log(x ÷ y). The quotient law is about division inside the log. A ratio of two separate logs is the change of base formula, which is a different thing entirely.