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Logarithm Calculator

A logarithm (log) is the operation that finds the exponent to which a base must be raised to get a number — the inverse of exponentiation. It is used in many fields such as sound level (decibels), earthquake magnitude (Richter), and pH.

Enter the argument and the base to calculate the logarithm. Use base 10 for the common logarithm, or e (about 2.718) for the natural logarithm.

For example, with base 10 and argument 100, log₁₀100 = 2.

Logarithm Calculator Calculation Method

log_b x = ln x / ln b
The log value is 0 .

About This Calculator

A logarithm answers the question "to what power must the base be raised to get this number?". It is the inverse of exponentiation, and because it turns multiplication into addition it was the original tool for fast calculation.

Enter the argument and the base — 10, e or any positive value — to calculate the logarithm.

A logarithm answers the question of which power produces a number, so log₁₀(1000) = 3 and log₂(1024) = 10. Products become sums through log(ab) = log a + log b, the identity that turned multiplication into addition for four centuries of calculation. Natural logarithms use base e ≈ 2.71828 and are the default in calculus and most programming languages.

Key Formula:log_b(x) = y ⟺ b^y = x

Frequently Asked Questions

What is the difference between log and ln?

"log" usually means base 10 (common logarithm) and "ln" means base e ≈ 2.71828 (natural logarithm). In science and finance ln is more common because of its calculus properties.

Why are logarithms used for earthquakes and sound?

The Richter scale and decibels are logarithmic because the underlying energies vary over enormous ranges. Each whole number on the Richter scale means about 32 times more energy released, which is far easier to write than the raw figures.

Why can I not take the logarithm of a negative number?

Any positive base raised to a real power gives a positive result, so no exponent maps to a negative number. Complex logarithms exist, but a real-valued calculator correctly reports an error.

What does a logarithmic scale make visible?

Equal steps multiply rather than add, so a chart can show 1 and 1,000,000 on the same axis. Earthquake magnitudes, decibels and pH are all logarithmic and use that property deliberately.

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Sources & last update

2026-10-19