---
title: "Straightening Data with Logarithmic Graphs"
description: "Take logarithms of a model to make it linear: y = axⁿ gives a straight line for log y against log x, and y = abˣ gives one for log y against x. The gradient and intercept then return the constants."
canonical: https://lightmysky.com/learn/mathematics/straightening-data-with-logarithmic-graphs-mt_TVar2jxRMI
source: https://lightmysky.com/learn/mathematics/straightening-data-with-logarithmic-graphs-mt_TVar2jxRMI.md
retrieved: 2026-09-12
---

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# Straightening Data with Logarithmic Graphs

Take logarithms of a model to make it linear: y = axⁿ gives a straight line for log y against log x, and y = abˣ gives one for log y against x. The gradient and intercept then return the constants.

Subject: Mathematics · Area: Algebra · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/straightening-data-with-logarithmic-graphs-mt_TVar2jxRMI

## Ready when they can

- Choose the axes that straighten a given model
- Recover a and n from the gradient and intercept of a log-log plot
- Say what a straight log-linear plot tells you about the data

## Lesson: Straight lines from log plots

Some curves turn straight when you plot their logs. If data follow y = a x^n, take logs to get log y = log a + n log x. Plot log y against log x and you get a line with gradient n and intercept log a. Plain y against x stays curved, so it cannot straighten a power law.

**Example.** Read a base 10 log-log plot this way. An intercept of 2 means log a = 2, so a = 10^2 = 100, and an intercept of 3 gives a = 1000. A line through (log x = 2, log y = 1) and (log x = 4, log y = 7) rises 6 over a run of 2, so the gradient is 3 and the power is 3. A gradient of 1/2 with intercept 0 means y = sqrt(x), since a = 1.

If data follow y = a b^x, log only the y values. Logs give log y = log a + x log b, which is a line in plain x with gradient log b and intercept log a. Read the intercept first to get a, then the gradient to get b. With base 10 logs a gradient of 2 means b = 10^2 = 100.

**Tip.** Let the slope tell the story before you compute. A rising log-linear line means growth with b above 1, while a falling one has a negative gradient, so log b is negative and b sits below 1. Changing a only shifts the intercept, never the gradient. Keep one base through the whole question and convert back with powers of 10.

**Recap.** Log both axes to read powers, log only y to read exponentials, and let gradient and intercept hand back the constants.

## Practice

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

## Needs first

- [Lines of Best Fit: Prediction and Its Limits](https://lightmysky.com/learn/mathematics/lines-of-best-fit-prediction-and-its-limits-mt_CnYCx1a0j5)
- [Solving Exponential Equations with Logarithms](https://lightmysky.com/learn/mathematics/solving-exponential-equations-with-logarithms-mt_DaxxpSbDOn)
- [Forms of the Equation of a Straight Line](https://lightmysky.com/learn/mathematics/forms-of-the-equation-of-a-straight-line-mt_dhIIxdFwcp)

## Opens up

- [Exploratory Analysis and an Honest Chart](https://lightmysky.com/learn/computing/exploratory-analysis-and-an-honest-chart-mt_Jpb78Z6Ytu)
- [The Arrhenius Equation and Finding Activation Energy](https://lightmysky.com/learn/science/the-arrhenius-equation-and-finding-activation-energy-mt_QBQDSNwbXK)