What a logarithm asks
log₂(32) asks "2 raised to what power gives 32?" Since 2⁵ = 32, the answer is 5. A logarithm is really the inverse question to exponentiation — instead of computing base^n, it finds n given the base and the result.
How it's actually computed: ln(value) ÷ ln(base)
Rather than searching for the right exponent by trial, logarithms are computed using the change-of-base formula: log_base(value) = ln(value) ÷ ln(base), using the natural logarithm (base e). For log₁₀(1000): ln(1000) ÷ ln(10) = 3, since 10³ = 1000.
Why this formula works for any base
The change-of-base formula means a calculator only ever needs to know how to compute the natural logarithm — every other base's logarithm is just a ratio of two natural logs. This is why log₂(32) = 5 and log₁₀(1000) = 3 can both be computed by the exact same underlying formula, just with different base values plugged in.
A special case, the logarithm of 1
For any base, the logarithm of 1 is always 0 — because any number raised to the power of 0 equals 1. ln(1), computed directly, is exactly 0, confirming this for the natural log base specifically.