Log Calculator

Find the logarithm of any positive number in any base, with the change-of-base steps shown so you can see exactly how the answer was reached.

Logarithm —

How to Calculate a Logarithm

Enter a value and a base to see the logarithm and the change-of-base working.

1

Enter your value and base

Type a positive number in the Value (x) field and a base in the Base field, like 1000 and 10. Use the ln or log₁₀ preset buttons to jump straight to base e or base 10, or type e in the base field for natural log.

2

The calculator applies change of base

It computes log_b(x) = ln(x) / ln(b), the change-of-base formula, so any base works — not just e and 10.

3

Read the result and steps

The logarithm appears at the top, and the ln(x), ln(base), and division steps are listed below it. Solving an equation instead? The Quadratic Formula Calculator handles quadratics with the same worked-steps format.

Frequently Asked Questions

What is a logarithm?
A logarithm answers the question "what power do I raise the base to, to get this number?" log_b(x) = y means b^y = x. For example, log_10(1000) = 3 because 10^3 = 1000. Logarithms are the inverse of exponentiation, and they turn multiplication into addition, which is why they show up in pH, decibels, the Richter scale, and computer-science time complexity.
What is the difference between ln and log?
"log" (without a subscript) conventionally means log base 10, while "ln" always means the natural log, log base e (e ≈ 2.71828). Base-10 logs are common in science and engineering for orders of magnitude; the natural log shows up naturally in calculus, growth and decay models, and compound interest. This calculator handles both — use the ln preset for base e or the log₁₀ preset for base 10, or type in any other base.
How does the change-of-base formula work?
Most calculators only have built-in keys for ln and log base 10, so to find a logarithm in any other base you use the change-of-base formula: log_b(x) = ln(x) / ln(b). For example, log_2(8) = ln(8) / ln(2) = 2.0794... / 0.6931... = 3. This calculator applies that formula for any base you enter and shows each intermediate value.
Why do logarithms only work for positive numbers?
Because log_b(x) = y means b^y = x, and a positive base raised to any real power always stays positive — it can never equal zero or a negative number. So there is no real exponent y that produces x ≤ 0, which is why this calculator (and every log function) requires x > 0. The base must also be positive and not equal to 1, since 1 raised to any power always stays 1.