---
title: "Logarithms as the Inverse of Exponentials"
description: "Read logₐ x as the answer to the question 'what power of a gives x'. The log graph is the exponential graph reflected in y = x, and each function undoes the other."
canonical: https://lightmysky.com/learn/mathematics/logarithms-as-the-inverse-of-exponentials-mt_skLQLUan0T
source: https://lightmysky.com/learn/mathematics/logarithms-as-the-inverse-of-exponentials-mt_skLQLUan0T.md
retrieved: 2026-09-12
---

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# Logarithms as the Inverse of Exponentials

Read logₐ x as the answer to the question 'what power of a gives x'. The log graph is the exponential graph reflected in y = x, and each function undoes the other.

Subject: Mathematics · Area: Algebra · Ages 16 to 17
Page: https://lightmysky.com/learn/mathematics/logarithms-as-the-inverse-of-exponentials-mt_skLQLUan0T

## Ready when they can

- Rewrite 2⁵ = 32 in logarithm form and back again
- Explain why the logarithm of a negative number has no value
- Sketch y = ln x by reflecting y = eˣ in the line y = x

## Lesson: Logs undo powers

Read log base a of x as a question: what power of a gives x. So log base 2 of 32 is 5, because 2 to the 5 is 32. Power form and log form are the same fact twice. Fluency both ways turns log equations into plain power questions.

Some questions have no answer. The log of zero or a negative has no value, because no power of a positive base lands there. Since e to the x only outputs positives, ln x only accepts positives. Domain limits are built in.

**Example.** Sketch y equals ln x by reflecting y equals e to the x in the line y equals x. Where the exponential crosses (0, 1), the log crosses (1, 0). Where the exponential soars, the log climbs slowly. The curve hugs the y axis, showing the domain edge.

**Tip.** Exponentials and logs undo each other by swapping inputs and outputs. One takes a power and gives a number, the other returns the power. That swap is exactly why their graphs mirror. Name which direction each function runs before solving.

**Recap.** A log names the missing power, accepts positives only, and mirrors its exponential.

## Practice

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

## Needs first

- [The Exponential Function and the Number e](https://lightmysky.com/learn/mathematics/the-exponential-function-and-the-number-e-mt_2hyjN4JfJJ)
- [Inverse Functions](https://lightmysky.com/learn/mathematics/inverse-functions-mt_7k7q5ZjMb4)

## Opens up

- [Variable Mass and the Rocket Equation](https://lightmysky.com/learn/science/variable-mass-and-the-rocket-equation-mt_4IugclcEVi)
- [The Laws of Logarithms](https://lightmysky.com/learn/mathematics/the-laws-of-logarithms-mt_5vHdWlxaqY)
- [Hyperbolic Functions and Their Inverses](https://lightmysky.com/learn/mathematics/hyperbolic-functions-and-their-inverses-mt_gCUl1sWi9M)
- [pH, Kw and Strong Acids and Bases](https://lightmysky.com/learn/science/ph-kw-and-strong-acids-and-bases-mt_KJ4yc25Z_l)