Logarithm Calculator

What is a logarithm calculator?

A logarithm calculator finds the power a given base must be raised to in order to produce a number — for example, log››10(1000) = 3 because 10³ = 1000. It works with any base, including the natural log and base 2.

How this calculator works

• Enter a positive number and a base (any base other than 1 or 0).
• It computes the log with the change-of-base formula: log_b(x) = ln(x) / ln(b).
• It also reports the natural log and the base-10 log of the same number.
• The antilog check raises the base back to the result to confirm the answer.

How to use the result

1. Use base-10 logs to compress wide-ranging data — pH, decibels, and earthquake magnitude are all logarithmic scales.
2. Use natural logs for exponential growth problems like compound interest and population models.
3. Use base-2 logs in computer science for binary search depth, tree height, and information theory.
4. Verify the antilog: if base^result returns your number, the log is correct.

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) = 2.079 / 0.693 = 3. This is why one logarithm button on a calculator is enough — any base reduces to a ratio of natural logs.

Common mistakes

• Taking the log of zero or a negative. Logs are undefined there — the argument must be positive.
• Using base 1 or 0. Every power of 1 is 1, so log base 1 has no meaningful answer.
• Confusing ln with log››10. They differ by a constant factor (~2.303), so mixing them shifts answers.

Logs turn multiplication into addition — the same way percentages turn absolute changes into comparable relative ones. The percentage calculator handles those relative comparisons, and the scientific notation converter is a log-friendly companion for extreme magnitudes.