Logarithm Calculator

Step 1: Enter the value (must be positive).

Step 2: Enter the base — use 10 for common log, 2.71828 for natural log (ln), or 2 for binary logarithm.

Step 3: Click Calculate. The result shows the log in your chosen base, plus ln and log₁₀ for comparison.

A logarithm answers the question: 'What exponent do I need to apply to the base to get this value?' For example, log base 10 of 1,000 equals 3, because 10 to the 3rd power equals 1,000.

Logarithms are the inverse operation of exponents. Wherever exponents compress repeated multiplication into a simpler notation, logarithms reverse the operation to extract the exponent.

The three most common logarithms are:
• Common log (log₁₀): base 10 — used in decibels, pH, Richter scale.
• Natural log (ln): base e ≈ 2.71828 — used in calculus and continuous growth.
• Binary log (log₂): base 2 — used in computer science and information theory.

Definition: log_b(x) = y means b^y = x

Change of base: log_b(x) = ln(x) / ln(b)

Product: log(a × b) = log(a) + log(b)
Quotient: log(a / b) = log(a) − log(b)
Power: log(aⁿ) = n × log(a)
Identity: log_b(b) = 1, log_b(1) = 0

Example: log₁₀(1000) = 3
Because 10³ = 1,000

Example 1: log₁₀(1,000,000) = 6 — there are six zeros, so the exponent is 6.

Example 2: ln(2.71828) = 1 — the natural log of e always equals 1.

Example 3: log₂(256) = 8 — in computing, 256 represents 8 bits of information.

Only positive values. Logarithms of zero or negative numbers are undefined in real numbers.

Base cannot be 1. log base 1 of anything is undefined (except log₁(1) which is indeterminate).

Decibels use log₁₀: a 10 dB increase is 10× more power.

pH scale uses −log₁₀ of hydrogen ion concentration.