Logarithm Calculator

Calculate the logarithm of any positive number with any positive base. Supports common log (base 10), natural log (base e), and binary log (base 2).

Base presets

Base

Value (x)

How It Works

The logarithm answers the question: "To what power must the base be raised to produce a given number?" If b^y = x, then log_b(x) = y.

For example, log₁₀(1000) = 3 because 10³ = 1000.

Common log (log₁₀) — Base 10, widely used in science and engineering
Natural log (ln) — Base e ≈ 2.71828, fundamental in calculus and physics
Binary log (log₂) — Base 2, essential in computer science and information theory

Any logarithm can be computed using the change of base formula:

log_b(x) = ln(x) / ln(b) = log(x) / log(b)

log_b(1) = 0

log_b(b) = 1

log_b(x × y) = log_b(x) + log_b(y)

log_b(x / y) = log_b(x) − log_b(y)

log_b(xⁿ) = n × log_b(x)

b^log_b(x) = x

x	log₁₀(x)	ln(x)	log₂(x)
0.01	−2	−4.6052	−6.6439
0.1	−1	−2.3026	−3.3219
1	0	0	0
2	0.3010	0.6931	1
e	0.4343	1	1.4427
10	1	2.3026	3.3219
100	2	4.6052	6.6439
1000	3	6.9078	9.9658

Decibels — Sound intensity measured on a logarithmic scale (base 10)
pH Scale — Acidity measured as −log₁₀ of hydrogen ion concentration
Richter Scale — Earthquake magnitude uses base-10 logarithms
Information Theory — Entropy and data compression use base-2 logarithms (bits)
Compound Interest — Time to double an investment: t = ln(2) / ln(1 + r)
Algorithm Analysis — Binary search and balanced trees have O(log n) complexity

Help us improve this page by providing feedback, and include your name/email if you want us to reach back. Thank you in advance.

Share with