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Logarithm Calculator

Computes the logarithm of a positive number to any base using the change-of-base formula, and always reports the natural, common (base 10) and binary (base 2) logarithms.

When to use

You need log_b(x) for an arbitrary base, or ln, log10 and log2 of a number at once.

Do not use when: You need to raise a base to a power (use exponent) or a root (use nth-root); for pH-style −log10 of a concentration use ph.

Formula

log_b(x) = ln(x) / ln(b) (change of base); ln = log_e, log10 = log_10, log2 = log_2

Inputs

Parameter	Type	Unit	Required	Description
`value`	number		yes	Positive number whose logarithm is taken. Range: > 0
`base`	number		default 10	Logarithm base; positive and not 1. Use 2.718281828459045 for the natural logarithm, or read the ln output. Range: > 0

Outputs

Output	Type	Description
`log`	number	Logarithm of value to the given base.
`ln`	number	Natural logarithm (base e).
`log10`	number	Common logarithm (base 10).
`log2`	number	Binary logarithm (base 2).
`expression`	string	The evaluated logarithm as text.

Example

log10 of 1000: {"value":1000,"base":10} → {"log":3,"ln":6.907755,"log10":3,"log2":9.965784,"expression":"log_10(1000) = 3"}

log2 of 8: {"value":8,"base":2} → {"log":3,"ln":2.079442,"log10":0.90309,"log2":3}

GET https://tttkmbb.com/api/v1/calculate/logarithm?value=1000&base=10

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/logarithm (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/logarithm · Markdown: https://tttkmbb.com/math/logarithm.md · JSON definition: https://tttkmbb.com/math/logarithm.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="logarithm"
OpenAPI operationId: calculate_logarithm
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Logarithm (reference)
NIST Digital Library of Mathematical Functions §4.2 – Elementary functions: definitions of the logarithm (standard)
Wolfram MathWorld – Logarithm (reference)

FAQ

Why can't the value be 0 or negative?

Real logarithms are only defined for positive numbers: log(0) tends to −∞ and logarithms of negatives are complex.

Which base does 'log' mean?

In science and calculators 'log' usually means base 10, in mathematics often base e, and in computing base 2. All three are reported so no convention is assumed.

Related calculators

Exponent Calculator — Inverse operation: b^y = x.
Nth Root Calculator — Roots as fractional exponents.
pH Calculator — pH is −log10 of hydrogen-ion concentration.