Math · Algebra
Logarithm calculator.
Enter a base and a number. You get the logarithm, the natural, common and binary logs, the change-of-base steps, and an optional check of the log laws.
log₂(8) = 3, because 2³ = 8. With y = 4 and a power of 3, the laws check out in base 2: log(8 × 4) = 5 = 3 + 2, log(8 / 4) = 1 = 3 − 2, and log(8³) = 9 = 3 × 3.
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The formulas
The logarithm log_b(x) is the exponent you put on b to get x, so log_b(x) = y means bʸ = x. Any calculator that only has natural log can still compute any base with the change-of-base formula log_b(x) = ln(x) / ln(b). The laws are log(xy) = log x + log y (product), log(x/y) = log x − log y (quotient) and log(xᵖ) = p log x (power), for positive x and y.
Worked example
log₂(8) = ln 8 / ln 2 = 2.07944154 / 0.69314718 = 3. Natural log of 10 (base e) is 2.30258509, and log₁₀(5000) = 3.69897. For a base-5 logarithm of 0.04 the answer is −2, since 5⁻² = 0.04.
Limits
Only real logarithms are computed, so x must be greater than 0 and the base must be positive and not 1. The log-law check compares both sides in floating-point arithmetic and reports them equal when they agree to about nine digits.
Sources, opened 15 October 2026: OpenStax, Algebra and Trigonometry 2e, 6.5 Logarithmic Properties (product, quotient and power rules; change-of-base formula log_b M = ln M / ln b).
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 log₂(8) that is 2.07944154 / 0.69314718 = 3.
What is the difference between log and ln?
ln is the natural logarithm with base e (about 2.71828), and log usually means base 10. ln(10) is 2.30258509 and log₁₀(10) is 1.
Can you take the log of a negative number or zero?
Not as a real number. A logarithm needs a positive input, so the calculator asks for x greater than 0.
What are the main logarithm laws?
Product: log(xy) = log x + log y. Quotient: log(x/y) = log x − log y. Power: log(xᵖ) = p × log x. They hold for any valid base.
Why can the base not be 1?
Every power of 1 is 1, so no exponent could give any other number. The change-of-base formula would also divide by ln 1 = 0.
Learning aid
Educational calculator with worked steps. Check answers and the required method against your own course or textbook. Nothing you type leaves your browser.