--- title: "The Laws of Logarithms" description: "Use matching logarithm bases and positive arguments to turn products into sums, quotients into differences and powers into multipliers. Expand or combine logarithms, and use change of base to evaluate" 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The Laws of Logarithms Use matching logarithm bases and positive arguments to turn products into sums, quotients into differences and powers into multipliers. Expand or combine logarithms, and use change of base to evaluate them. 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 for positive x and y - Expand log(a²b / c) into separate terms for positive a, b and c - Use a change of base to evaluate a logarithm the calculator has no key for ## Lesson: Combining logs with the three laws You have used logs to name exponents. Here log means base 10; ln means base e. A log base must be positive and not 1, and its argument must be positive. The product and quotient laws combine logs with the same base. Expanding log(x^2) into 2 log x requires x > 0: a negative x has a positive square, but log x has no real value. For positive M and N, log(MN) = log M + log N and log(M/N) = log M - log N. Also log(M^r) = r log M for any real r. Read these laws backwards to condense. Move coefficients into powers first, then multiply or divide the arguments. Each subtracted log gives a denominator factor. For positive s, t and u, 3 log s + log t - 2 log u = log(s^3 t/u^2). *(drawing: The coefficient 3 becomes an exponent. Adding the logs then multiplies their arguments.)* **Example.** To expand, separate products and quotients, then bring exponents to the front. For positive r, s and t, log(r^3 s/t^2) = log(r^3) + log s - log(t^2) = 3 log r + log s - 2 log t. A power of a product affects both factors: log((rs)^3) = 3(log r + log s). If a denominator contains a product, subtract its whole log expansion. The product law applies to multiplication inside a log. It does not split terms joined by + inside the argument. For example, log(r+s) must stay together. Check a proposed identity with positive numbers: log(1+1) = log 2 > 0, while log 1 + log 1 = 0. One counterexample is enough to show that a claim does not hold for every allowed value. Change of base divides the log of the argument by the log of the original base: log base b of M = log(M)/log(b) = ln(M)/ln(b). Here M > 0, b > 0 and b != 1. Both logs in a fraction use the same new base. For example, log base 5 of 125 = log(125)/log(5). Since 125 = 5^3, the power law gives 3 log(5)/log(5) = 3. *(drawing: The original base goes in the denominator. The power law gives an exact value without rounding.)* **Example.** If the argument and base are powers of one number, apply the power law to both logs. For example, log base 8 of 16 = log(2^4)/log(2^3) = 4 log(2)/(3 log(2)) = 4/3. You can also divide given log values. If log base 3 of A = 7 and log base 3 of B = 4, then A = 3^7 and B = 3^4. Changing to base 3 gives log base B of A = 7/4. **Recap.** Use matching bases and positive arguments. Products give sums, quotients give differences, and powers give multipliers. Change base by dividing by the log of the original base. *Lesson, questions and flashcards written for this dataset.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Logarithms as the Inverse of Exponentials](https://lightmysky.com/learn/mathematics/logarithms-as-the-inverse-of-exponentials-mt_skLQLUan0T) ## Optional connections - [Zero and Negative Indices](https://lightmysky.com/learn/mathematics/zero-and-negative-indices-mt_jAtXBSVqkj) ## 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)