ToolForage.
MATH CALCULATOR

Logarithm Calculator

Find the exponent that produces a number in a chosen base. Select common log, natural log, binary log, or a custom base.

Calculator

Must be positive and different from 1.
YOUR RESULTS
Logarithm3
Base used
10
Equivalent power statement
10 ^ 3 ≈ 1000

The logarithm is the exponent of the selected base. The power statement uses a rounded exponent.

The exponent represented by a logarithm

A logarithm answers “which exponent?” The equivalent power statement connects the result to exponentiation. Results use ten displayed decimal places when needed, rather than claiming an exact symbolic form.

To calculate a power from a known base and exponent, use the Exponent Calculator .

The formula

logᵦ(x) = ln(x) / ln(b), where x > 0, b > 0, and b ≠ 1.

The change-of-base formula divides two natural logarithms. Bases between zero and one are valid and reverse the direction of growth. A base of one is excluded because every power of one equals one, so it cannot define a logarithm function.

Worked example

With x = 1,000 and base 10, the logarithm is 3 because 10³ = 1,000. With x = 8 and base 2, the result is 3. With x = 4 and base 0.5, the result is −2.

How to use this calculator

  1. Enter a strictly positive number.
  2. Choose the base; enter a custom value only when Custom base is selected.
  3. Calculate and compare the result with the equivalent power statement.

Common log, natural log, and binary log

These choices share the same idea but measure powers of different bases. Common log measures powers of ten and is useful for orders of magnitude. Natural log uses e, approximately 2.71828. Binary log measures powers of two. A value can therefore have three different logarithms without any contradiction.

A negative result is possible

The number inside the logarithm must be positive, but the result may be negative. In a base greater than one, a number between zero and one has a negative logarithm. For example, log₁₀(0.01) = −2. When the base is between zero and one, that sign pattern reverses.

Precision near a base of one

A base close to one has a very small natural logarithm. Dividing by it can magnify input rounding and produce a large result. Check the entered digits before using such an answer in another calculation. This tool does not simplify symbolic logarithms or evaluate logarithms of complex numbers.

Assumptions & limitations

What this calculation assumes

  • Inputs represent positive real numbers.

What to keep in mind

  • Custom bases equal to 1 are invalid; ordinary floating-point precision applies.

Common questions

What is the logarithm of 1?

It is zero in every valid base, because b⁰ = 1.

Is ln different from log?

ln specifically means base e. The label log can mean different bases in different contexts; this calculator names the selected base explicitly.

Sources & further reading