Logarithm Calculator

The logarithm calculator finds the logarithm to any base, the natural logarithm ln and the common logarithm log (base 10). If the number is a power of the base, the calculator finds the exact value (for example, log₈ 32 = 5/3); otherwise it gives an approximation with the change of base formula. You can enter the base and the number as fractions, roots and powers: 1/8, √2, 2^5, e^3.

What is a logarithm

The logarithm of a number b to the base a is the exponent to which a must be raised to get b:

log_a b = c   ⇔   a^c = b

A logarithm is defined for a > 0, a ≠ 1 and b > 0. For example, log₂ 8 = 3 because 2³ = 8. The logarithm to base 10 is called the common logarithm and written log b, the logarithm to base e ≈ 2.71828 is the natural logarithm ln b.

Properties of logarithms

How to find a logarithm exactly

Write the base and the number as powers of the same number, then take the exponents out using the power rules:

Example 1. log₈ 32: 8 = 2³, 32 = 2⁵.
log_2³ 2⁵ = 5/3 · log₂ 2 = 5/3

Example 2. log_1/3 27: 27 = (1/3)⁻³, so log_1/3 27 = −3.

Example 3. log 0.001 = log 10⁻³ = −3,   ln √e = ln e^1/2 = 1/2.

Approximate value

If the number is not an exact power of the base, the logarithm is irrational. It is found with the change of base formula to natural logarithms:

Example 4. log₂ 10 = ln 10 / ln 2 ≈ 2.302585 / 0.693147 ≈ 3.321928

The logarithm of zero or a negative number does not exist among real numbers: a positive base raised to any power gives a positive number.