Calculate natural logarithms (ln), decimal (log₁₀) and any custom base.
A logarithm answers the question 'to what exponent must I raise the base to get this number?' Formally, log_b(x) = y means b^y = x, so the logarithm is the inverse of exponentiation. The natural logarithm ln uses base e ≈ 2.71828, the common logarithm log₁₀ uses base 10, and this tool also accepts any custom base b, as long as b > 0 and b ≠ 1. Internally every case is evaluated with the change-of-base formula log_b(x) = ln(x) / ln(b), which is why arbitrary bases work from a single built-in ln. The value x must be positive, because powers of a positive base are always positive; the tool also highlights key properties such as log_b(1) = 0 and log_b(b) = 1, and shows the antilogarithm for checking.
By definition, log_b(x) is the exponent y such that b^y = x. The three most common bases have their own names: log₁₀ (common), ln (natural, base e), and log₂ (binary). For any base, the tool uses the change-of-base formula log_b(x) = ln(x) / ln(b), which reduces every calculation to a single natural logarithm.
The operating properties turn multiplications into additions and powers into products, which historically made logarithms essential for hand computation: log(x·y) = log(x) + log(y); log(x/y) = log(x) − log(y); log(xⁿ) = n·log(x).
Example 1, decimal: log₁₀(1000) = 3, because 10³ = 1000. Via change of base, that is ln(1000) / ln(10) = 6.90776 / 2.30259 = 3.
Example 2, base 2: log₂(50) = ln(50) / ln(2) = 3.91202 / 0.69315 ≈ 5.643856. Checking, 2^5.643856 ≈ 50.
The real domain of the logarithm is x > 0: for x = 1 the result is 0, and when 0 < x < 1 (with base > 1) the logarithm is negative, tending to −∞ as x approaches zero. The base must be positive and different from 1, because with base 1 the equation 1^y = x has no unique solution.
Logarithmic scales measure quantities that vary across orders of magnitude: pH in chemistry, decibels in sound, the Richter earthquake scale, information entropy (base 2), and the complexity of algorithms such as binary search (O(log n)).
Paste the code into your HTML and the tool shows up on your page, without J-Kit's navigation and ads. It still runs in the browser of whoever visits your site.
<iframe
src="https://jkit.tools/embed/en-US/logarithm-calculator"
width="100%"
height="600"
style="border:0"
loading="lazy"
title="Logarithm Calculator"
></iframe>These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.
log_b(x) = y means b^y = x, so a logarithm is simply an exponent, the power the base must be raised to. It is the inverse operation of exponentiation. For example, log₂(8) = 3 because 2³ = 8, and log₁₀(100) = 2 because 10² = 100.
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