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

Understanding Logarithms

A logarithm answers the question: to what power must the base be raised to get a given number? log₁₀(1000) = 3 because 10³ = 1000. The natural log (ln) uses base e ≈ 2.718, appearing in continuous growth and decay equations. The binary log (log₂) is used in computer science to measure information and algorithmic complexity. Change-of-base: log_b(x) = log(x)/log(b), letting any calculator compute any base.

Logarithms compress large ranges: the Richter scale, decibel scale, pH scale, and stellar magnitude scale are all logarithmic. A magnitude-7 earthquake is 10× stronger than magnitude-6, not just 1 unit stronger. pH 4 is 10× more acidic than pH 5. Logarithms also solve exponential equations (compound interest, population growth, radioactive decay) and appear in entropy, information theory, and machine learning loss functions.

Key Logarithm Values

Expression	Result	Why It Matters
log₁₀(1)	0	Any base: logₐ(1)=0
log₁₀(10)	1	Base matches argument
log₁₀(100)	2	10²=100
log₂(8)	3	2³=8, used in CS
ln(e)	1	Natural log of e
ln(1)	0	ln(1) always = 0

Logarithm Laws and Their Applications

Logarithms obey a set of algebraic laws that simplify complex calculations. The product rule states that the logarithm of a product equals the sum of the logarithms of its factors. The quotient rule states that the logarithm of a quotient equals the difference of the logarithms. The power rule states that the logarithm of a number raised to an exponent equals the exponent multiplied by the logarithm of the base number. These three rules, combined with the change-of-base formula (which expresses any logarithm in terms of natural or common logarithms), allow evaluation of logarithms with arbitrary bases using any scientific calculator. Logarithms appear throughout science and engineering: the Richter scale measures earthquake magnitude logarithmically, decibels measure sound intensity logarithmically, pH measures acidity logarithmically, and information entropy in computer science is calculated using the binary logarithm (base 2). Signal processing, data compression, and machine learning algorithms all rely on logarithmic mathematics. Compound interest calculations use the natural logarithm to find the time required for an investment to reach a target value at a given continuous growth rate.

Logarithm Laws Reference Table

Law	Formula	Example (base 10)
Product rule	log(ab) = log a + log b	log(100×10) = 2 + 1 = 3
Quotient rule	log(a/b) = log a − log b	log(1000/10) = 3 − 1 = 2
Power rule	log(aⁿ) = n × log a	log(10²) = 2 × 1 = 2
Change of base	log_b(x) = ln(x)/ln(b)	log₂(8) = ln8/ln2 = 3
Identity	log_b(b) = 1	log₁₀(10) = 1
Zero rule	log_b(1) = 0	log₁₀(1) = 0