Logarithms & the Laws of Logs

A logarithm is just an exponent wearing a disguise. It answers one question — “what power?” — and once you hear it that way, the log laws stop being rules to memorise and start looking like obvious consequences of how powers multiply.

What a logarithm really asks

Since 2⁵ = 32, we say that log₂ 32 = 5. The logarithm simply reads off the power you need. In words, “log base 2 of 32” asks “2 to what power gives 32?”— and the answer is 5.

That makes a logarithm the inverse of an exponential. If bʸ = x, then log_b x = y, and the two statements say exactly the same thing from opposite ends. Raising to a power and taking a log undo each other, the way squaring and square-rooting do.

A logarithm hands back the exponent. Ask it “what power?” and it tells you.

Watch multiplication turn into addition

The three laws below all flow from a single fact: multiplying powers of the same base adds their exponents. Because a logarithm is an exponent, every product tucked inside a log becomes a sum outside it.

Step through each line and read the note that explains the move.

The three log laws

Each law rewrites an operation on the inside of a log as a simpler operation on the outside. They hold for any base, provided it is the same throughout, so we write log without a base below.

Product law

log(AB) = log A + log B. A product inside becomes a sum outside. It follows straight from the index law bᵐ × bⁿ = bᵐ⁺ⁿ: multiplying the numbers adds their exponents, and those exponents are exactly what the logs report.

Quotient law

log(A/B) = log A − log B. Dividing inside becomes subtracting outside, mirroring bᵐ ÷ bⁿ = bᵐ⁻ⁿ.

Power law

log(Aⁿ) = n log A. A power inside drops down to the front as a multiplier — because (bᵐ)ⁿ = bᵐⁿ. This is the law that lets you solve equations where the unknown is stuck in the exponent.

Worked examples

Common mistakes

Practice