---
title: "The Laws of Logarithms"
description: "Turn products into sums, quotients into differences and powers into multipliers, and combine several logarithms into one before solving or simplifying."
canonical: https://lightmysky.com/learn/mathematics/the-laws-of-logarithms-mt_5vHdWlxaqY
source: https://lightmysky.com/learn/mathematics/the-laws-of-logarithms-mt_5vHdWlxaqY.md
retrieved: 2026-09-12
---

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# The Laws of Logarithms

Turn products into sums, quotients into differences and powers into multipliers, and combine several logarithms into one before solving or simplifying.

Subject: Mathematics · Area: Algebra · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/the-laws-of-logarithms-mt_5vHdWlxaqY

## Ready when they can

- Write log x + 2 log y as a single logarithm
- Expand log(a²b / c) into separate terms
- Use a change of base to evaluate a logarithm the calculator has no key for

## Lesson: Combining logs with the three laws

You already know a log asks which power gives a number. The three laws all follow from that idea. Adding two logs means multiplying what sits inside them. Subtracting means dividing what sits inside. A multiplier in front dives inside as a power.

**Example.** Take log x + 2 log y. Pull the 2 inside first, so 2 log y becomes log(y^2). Then the plus fuses the pair into log(x y^2). The same two moves turn 2 log x - log y into log(x^2/y).

**Example.** Take log(a^2 b/c). Split the division first to get log(a^2 b) - log c. Split the product next to get log(a^2) + log b - log c. Bring the power down last to get 2 log a + log b - log c. Read backwards, the same laws condense: 2 log a + log b - log c fuses into log(a^2 b/c).

**Tip.** Some bases have no calculator key, so rewrite with base 10 logs. Log base 2 of 8 becomes log(8)/log(2), and since 2^3 = 8 the value is 3. Log base 4 of 8 becomes log(8)/log(4), which is 1.5, since 4^1.5 = 8. Pick one base and stay with it for the whole question.

**Recap.** Pull powers inside first, then turn plus into multiply and minus into divide, and change base with a division of logs.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Zero and Negative Indices](https://lightmysky.com/learn/mathematics/zero-and-negative-indices-mt_jAtXBSVqkj)
- [Logarithms as the Inverse of Exponentials](https://lightmysky.com/learn/mathematics/logarithms-as-the-inverse-of-exponentials-mt_skLQLUan0T)

## Opens up

- [Solving Exponential Equations with Logarithms](https://lightmysky.com/learn/mathematics/solving-exponential-equations-with-logarithms-mt_DaxxpSbDOn)
- [Phase Diagrams, the Clapeyron Equation and the Phase Rule](https://lightmysky.com/learn/science/phase-diagrams-the-clapeyron-equation-and-the-phase-rule-mt_U-zj6r4WFM)