How do you calculate a logarithm with any base?

Use the change-of-base formula: log_b(x) = ln(x) / ln(b). For example, log_2(8) = ln(8)/ln(2) = 3.

Logarithm Calculator

Calculate the logarithm of any number with any base, including log base 2, 10, and e.

By AbraCalc Editorial Team · Reviewed by AbraCalc Methodology Review · Last reviewed: 2026-06-25 1 source

Embed this tool on your site

<iframe src="https://abracalc.com/calculator/logarithm-calculator/embed.html" width="340" height="320" style="border:1px solid #e6e1f0;border-radius:12px" loading="lazy"></iframe>

Cite this tool

APA

AbraCalc. (2026). Logarithm Calculator [Online calculator]. Retrieved from https://abracalc.com/calculator/logarithm-calculator/

BibTeX

@misc{abracalc-logarithm-calculator, author = {AbraCalc}, title = {Logarithm Calculator}, year = {2026}, howpublished = {\url{https://abracalc.com/calculator/logarithm-calculator/}} }

Did this tool answer your question?

How to use this tool

Enter number and base in the fields above.
Results update instantly as you type — or click Calculate.
Read your log result and the full breakdown beneath it.

Calculate log_base(x) using the change-of-base formula: ln(x)/ln(base).

Formula

log_base(x) = ln(x) / ln(base)

This uses the change-of-base identity, where ln denotes the natural logarithm.

How it works

The calculator applies the change-of-base formula, dividing the natural logarithm of x by the natural logarithm of the chosen base to obtain the logarithm in any base. Both x and the base must be positive and the base must not equal 1; entering invalid values will produce NaN or ±Infinity.

Worked example

Worked example: log base 2 of 8

Input: x = 8, base = 2.
Apply change-of-base: ln(8) / ln(2).
ln(8) ≈ 2.0794; ln(2) ≈ 0.6931.
Divide: 2.0794 / 0.6931 = 3.

log₂(8) = 3

Common mistakes to avoid

Confusing log base 10 (common log) with natural log (ln); entering log(100) and expecting ln(100) = 4.605 instead of 2.
Treating log(x) as defined for x <= 0; the logarithm is undefined for non-positive inputs.
Forgetting to apply the change-of-base formula when using a non-standard base, e.g. computing log₂(8) as log(8) instead of log(8)/log(2).

Key terms

Logarithm: The exponent to which a base must be raised to produce a given number; log_b(x) = y means b^y = x.
Natural logarithm (ln): The logarithm with base e (≈ 2.71828); the basis for the change-of-base formula.
Change-of-base formula: The identity log_b(x) = ln(x) / ln(b), used to compute logarithms in any base from natural logs.
Common logarithm (log₁₀): The base-10 logarithm, widely used in science and engineering; log₁₀(1000) = 3.
Binary logarithm (log₂): The base-2 logarithm, important in computer science and information theory.

Frequently asked questions

How do you calculate a logarithm with any base?: Use the change-of-base formula: log_b(x) = ln(x) / ln(b). For example, log_2(8) = ln(8)/ln(2) = 3.

References & sources

Wikipedia: Logarithm