Logarithm Base Converter
Convert log_a(x) into log_b(x) using change-of-base.
Overview
The Logarithm Base Converter rewrites a logarithm from one base to another via the change-of-base formula. Whether you have log_2(x) and need log_10(x), or ln(x) and want log_2(x), the converter gives you the equivalent value instantly.
It is built for computer-science students working with information-theory logs in bits, chemistry students switching between natural and common logs and engineers calibrating decibel scales. Most calculators only offer log (base 10) and ln (base e), leaving you to remember the conversion factor.
How it works
The change-of-base identity is log_b(x) = log_a(x) / log_a(b) for any positive base a. In particular log_b(x) = ln(x) / ln(b) = log10(x) / log10(b), so you can always go through natural or common logs.
This means once you have a logarithm in any one base, the value in any other base is just a constant multiple. For example log_2(x) = ln(x) / ln(2) ≈ ln(x) * 1.4427 — the factor 1 / ln(2) shows up in countless complexity-analysis formulas.
Examples
log_2(8) → 3 (since 2^3 = 8)
ln(10) → ≈ 2.3026, so log10(x) = ln(x) / 2.3026
log_5(125) using base 10 → log10(125) / log10(5) = 3
log_10(1000) using base e → ln(1000) / ln(10) = 3
FAQ
Why does the formula work?
Both sides equal the same exponent that turns the chosen base into x. Dividing by log_a(b) cancels out the choice of intermediate base.
Can the base be a fraction?
Yes, any positive base other than 1 works. A base less than 1 gives logarithms with flipped signs.
What if x is negative or zero?
The logarithm is undefined for non-positive x in real numbers. Complex logarithms exist but aren't covered here.
Why is base 2 important for computing?
Each bit doubles the number of representable values, so base-2 logs measure information content. Algorithm complexities like O(n log n) usually mean log base 2.
Are log and ln always different bases?
By convention, log without a base usually means base 10 (engineering) or base e (pure mathematics). ln always means base e to avoid ambiguity.