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The laws of logarithms — log(xy) = log x + log y, log(x/y) = log x − log y and log(xⁿ) = n log x — combine and simplify logarithmic expressions and are essential for solving equations with logs. It's a key topic in A-Level and International A-Level (iAL) Pure Mathematics. This free interactive eClassroom course walks through each step with worked examples and instant-feedback questions, so you can practise until it's secure.
How you do it
Adding logs multiplies the arguments; subtracting divides them; a coefficient in front becomes a power inside. Combine into a single log before undoing it. The laws never apply across a sum inside the log — that is the most common error.
Before you start
It assumes you can handle logarithm notation. This sits in Pure 2. Getting it right here pays off in solving equations and linear-form graphs.
What you learn
That single fact gives three laws — the product, quotient and power laws — that let you combine several logarithms into one, or split one into several. The laws only combine logs with the same base. Everything on the top of the fraction is added; everything on the bottom is subtracted. Chain the laws to reach an exact value, or to rewrite one logarithm using values you're given. The course works through the product and quotient laws, the power law, combining several terms into one logarithm, splitting one logarithm into separate terms and evaluating exactly, and “in terms of”. Every section has instant-feedback practice questions matched to Edexcel International A-Level and 9MA0 Pure, and the course finishes with full exam-style questions you mark yourself.
Related Topics
In our A-level maths courses, Laws of Logarithms belongs to Exponentials & Logarithms and is met in the AS year. The course is built from 5 concept sections, 5 worked examples and 5 check points before the exam-style set. Nearby topics in the same strand are Logarithmic Graphs (Linear Form), Natural Logarithms & e, Solving Equations Using Logarithms, Changing the Base and Exponential Functions & Graphs. The Exponentials & Logarithms revision engine gives you as many extra questions as you want, tracked on your skill tree.
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