A logarithm answers the question: to what power must a base be raised to produce a given number? Written log_b(x) = y, it means b^y = x. For example, log_10(1000) = 3 because 10^3 = 1000. The base must be a positive number other than 1. The most common bases are 10 (common logarithm, written 'log'), e (natural logarithm, written 'ln'), and 2 (binary logarithm, written 'lb' or 'log2').

How do you calculate a log in a different base?

Use the change-of-base formula: log_b(x) = ln(x) / ln(b), where ln is the natural logarithm. Most calculators only have log base 10 and ln built in, so this formula lets you calculate any base. For example, log_2(8) = ln(8) / ln(2) = 2.0794... / 0.6931... = 3, which confirms that 2^3 = 8.

What is the difference between log and ln?

'log' without a stated base almost always means log base 10 (the common logarithm) in everyday maths, engineering, and science. 'ln' always means the natural logarithm, which uses base e (approximately 2.71828). Mathematicians sometimes use 'log' to mean ln, but in calculator and computing contexts, log base 10 is the convention. Log base 2 is common in computer science and information theory.

Log Calculator

Calculate the logarithm of any positive number in any base. Choose log base 10 (common log), natural log (ln, base e), log base 2 (binary log), or enter a custom base. Results update instantly with step-by-step working shown.

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Standard Formula Change-of-base formula: logˣ(x) = ln(x) / ln(b). Verified against standard mathematical definitions.

1. Input

Number (x) must be positive

Logarithm base

Custom base positive, not 1

2. Notation

Decimal places in result

Worked example: log₁₀(1000) = 3 exactly, because 10³ = 1000.

Result

Expression

log₁₀(1000)

What you are calculating

Logarithm Result
3

Verification

10³ = 1000

base ^ result = x

Step-by-Step Working

Key Values

Number (x)1000

Base (b)10

ln(x)6.9078

ln(b)2.3026

logˣ(x) = ln(x) / ln(b)3

Related Logarithms

log₁₀(x)3.0000

ln(x)6.9078

log₂(x)9.9658

logˣ(x) [your base]3.0000

What Is a Logarithm?

A logarithm is the inverse of exponentiation. Where an exponent says "what is b raised to the power y?", a logarithm asks "to what power must b be raised to produce x?" The relationship is:

logˣ(x) = y means bʸ = x

For example, log₁₀(1000) = 3 because 10³ = 1000. The number b is called the base, x is the argument (or antilogarithm), and y is the logarithm.

Common Logarithm Bases

Base	Symbol	Common Name	Used In
10	log or log₁₀	Common logarithm	Engineering, decibels, pH, Richter scale
e (2.71828...)	ln	Natural logarithm	Calculus, growth/decay, statistics
2	log₂ or lb	Binary logarithm	Computer science, information theory, bits

The Change-of-Base Formula

Most calculators only provide log base 10 and ln. To find a logarithm in any other base, use the change-of-base formula:

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

This works because if logˣ(x) = y, then bʸ = x. Taking the natural log of both sides gives y · ln(b) = ln(x), so y = ln(x) / ln(b).

Example: log₂(32) = ln(32) / ln(2) = 3.4657... / 0.6931... = 5, confirming that 2⁵ = 32.

Logarithm Rules

Rule	Formula	Example
Product rule	logˣ(xy) = logˣ(x) + logˣ(y)	log(100 × 10) = log(100) + log(10) = 2 + 1 = 3
Quotient rule	logˣ(x/y) = logˣ(x) - logˣ(y)	log(100/10) = log(100) - log(10) = 2 - 1 = 1
Power rule	logˣ(xˣ) = p · logˣ(x)	log(10³) = 3 · log(10) = 3
Log of 1	logˣ(1) = 0	Any base raised to 0 = 1
Log of base	logˣ(b) = 1	log₁₀(10) = 1
Change of base	logˣ(x) = ln(x) / ln(b)	log₂(8) = ln(8)/ln(2) = 3

Worked Example

Default inputs: x = 1000, base = 10.

We want log₁₀(1000).
Apply the change-of-base formula: ln(1000) / ln(10).
ln(1000) = 6.907755... and ln(10) = 2.302585...
6.907755 / 2.302585 = 3.0000 exactly.
Verification: 10³ = 1000. Correct.

Related Calculators

Maths and Stats Calculators: full list of mathematical tools.
Antilog Calculator: the inverse of the logarithm (bʸ = x).
Exponent Calculator: calculate b raised to the power y.
Scientific Notation Calculator: convert numbers to and from scientific notation.
Quadratic Formula Solver: solve quadratic equations step by step.

Sources and method: Change-of-base formula: logˣ(x) = ln(x) / ln(b), derived from standard logarithm theory. Definitions from NIST Digital Library of Mathematical Functions (dlmf.nist.gov). JavaScript uses Math.log() (natural log) for all calculations.

This calculator computes logarithms using IEEE 754 double-precision floating-point arithmetic. Results are accurate to approximately 15 significant figures. For x very close to 0 or extremely large, rounding may affect the final displayed decimal places.

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