A logarithm answers the question: 'to what power must the base be raised to get this number?' log_b(x) = y means b^y = x. Example: log₁₀(1000) = 3 because 10³ = 1000. log₂(8) = 3 because 2³ = 8. The natural log ln(x) = log_e(x) where e ≈ 2.71828. Common log = log₁₀. Logarithms are the inverse of exponentiation.

What are the log rules I need to know?

Key logarithm rules: Product rule: log(a×b) = log(a) + log(b). Quotient rule: log(a/b) = log(a) − log(b). Power rule: log(a^n) = n×log(a). Change of base: log_b(x) = ln(x)/ln(b) = log(x)/log(b). Special values: log(1) = 0 (any base). log_b(b) = 1. ln(e) = 1.

What is the natural log (ln)?

The natural logarithm (ln) uses base e ≈ 2.71828 (Euler's number). ln(x) = log_e(x). It appears naturally in calculus, compound interest, population growth, and physics. The derivative of ln(x) is 1/x. Key values: ln(1) = 0; ln(e) = 1; ln(e²) = 2; ln(10) ≈ 2.303. To convert: ln(x) = log₁₀(x) × 2.302585.

Log Calculator

Enter a positive number to get log base 10, natural log (ln), log base 2, and any custom base.

Logarithm — Guide

What is a logarithm?

logᵉ(x) = y ⇔ b^y = x — "To what power must b be raised to get x?"

Key identities

log(a × b) = log a + log b • log(a / b) = log a − log b • log(aⁿ) = n × log a

Change of base

logᵉ(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)

Quick reference

x	log₁₀(x)	ln(x)	log₂(x)
1	0	0	0
2	0.301	0.693	1
10	1	2.303	3.322
100	2	4.605	6.644
1000	3	6.908	9.966

Related Calculators

Common Uses of Logarithms

Logarithms convert multiplication into addition and exponentiation into multiplication, which made them indispensable before calculators existed. Today they appear in: the Richter scale for earthquakes (each unit = 10× more energy), decibels for sound (a 10 dB increase = 10× louder), pH in chemistry (pH = −log[H⁺]), and information theory (Shannon entropy uses log₂). In finance, continuous compounding uses the natural logarithm: A = Pe^(rt), and ln is its inverse.

In machine learning, log-loss (cross-entropy) measures how well a probability model predicts outcomes. Log scales are also used in charts when data spans many orders of magnitude.

Logarithm Rules Quick Reference

Rule	Formula	Example
Product	log(ab) = log a + log b	log(100×10) = 2+1 = 3
Quotient	log(a/b) = log a − log b	log(1000/10) = 3−1 = 2
Power	log(aⁿ) = n·log a	log(10³) = 3·1 = 3
Change of base	log_b(x) = ln x / ln b	log₂(8) = ln8/ln2 = 3