--- title: "Solving Exponential Equations with Logarithms" description: "Take logarithms of both sides to bring an unknown down from an exponent, and solve equations that are quadratic in eˣ by substitution." canonical: https://lightmysky.com/learn/mathematics/solving-exponential-equations-with-logarithms-mt_DaxxpSbDOn source: https://lightmysky.com/learn/mathematics/solving-exponential-equations-with-logarithms-mt_DaxxpSbDOn.md retrieved: 2026-09-12 --- > **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`. # Solving Exponential Equations with Logarithms Take logarithms of both sides to bring an unknown down from an exponent, and solve equations that are quadratic in eˣ by substitution. Subject: Mathematics · Area: Algebra · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/solving-exponential-equations-with-logarithms-mt_DaxxpSbDOn ## Ready when they can - Solve 3ˣ = 20 to three significant figures - Solve e^(2x) - 5eˣ + 6 = 0 using a substitution - Find how long a modelled quantity takes to halve ## Lesson: Freeing x with logarithms You isolate the exponential part first, then take logs of both sides. The power rule pulls the exponent down front, and that log step is what frees x. Take 3^x = 20: logs give x log 3 = log 20, so x = log 20/log 3, about 2.73. Round only at the very end. When both sides share a base, skip the logs and match the exponents. Since 125 is 5^3, the equation 5^x = 125 gives x = 3 at once. Use this shortcut whenever the same base appears on both sides. **Example.** Treat e^(2x) - 5e^x + 6 = 0 as a quadratic in disguise. Substitute y = e^x to get y^2 - 5y + 6 = 0, which factors to y = 2 or y = 3. Undo the swap with logs: x = ln 2 or x = ln 3, not 2 or 3. **Example.** For halving stories, count how many half lives fit and halve that many times. A sample of 80 g with a 4 hour half life gives 3 half lives in 12 hours, so 80 to 40 to 20 to 10 g. A tracer falling from 200 to 25 MBq halves 3 times, and at 6 hours each that takes 18 hours. **Recap.** Free x with logs, swap e^x for y in quadratic cases, and count half lives in decay stories. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [The Laws of Logarithms](https://lightmysky.com/learn/mathematics/the-laws-of-logarithms-mt_5vHdWlxaqY) - [The Quadratic Formula](https://lightmysky.com/learn/mathematics/the-quadratic-formula-mt_C-4brnRYHY) ## Opens up - [Straightening Data with Logarithmic Graphs](https://lightmysky.com/learn/mathematics/straightening-data-with-logarithmic-graphs-mt_TVar2jxRMI)