log Logarithm Calculator

Calculate common log, natural log, and custom-base logarithms with change-of-base steps and inverse-power checks.

log, ln, and Custom Base — Inverse of Exponentiation

BrainyCalculators editorial insight — unique to this tool

log₁₀(1000) = 3; ln(e) = 1. pH = −log₁₀[H⁺] in chemistry; decibels use log scale for sound intensity. log₂ appears in algorithm complexity (O(n log n)) and information theory (entropy bits).

When to use this calculator

Use when solving equations or evaluating logs in any base. For powers directly, use Exponent.

Raising a number to a power instead?

This page finds logarithms. For power evaluation and scientific notation, use the Exponent Calculator →

Logarithm

Number (x)

Base

Antilogarithm

Enter a log value to find the original number (base^value).

Log value

Base

What is a Logarithm Calculator?

A logarithm calculator finds log₁₀, ln, or log_b(x) and shows change-of-base working. Logarithms answer “to what power must the base be raised to get x?”

Use this page for pH, decibels, compound growth inversion, and log-scale data. It is the inverse of exponentiation, not a general equation solver.

For b^n directly, use the Exponent Calculator. For polynomial or matrix work, use Algebra or Matrix tools instead.

Logarithm Formulas

Definition: logₐ(x) = y ⟺ aʸ = x

Change of base: logₐ(x) = ln(x) / ln(a) = log(x) / log(a)

Antilog: antilog(y) = 10ʸ | e-antilog(y) = eʸ

Worked Examples

Example — log₅(125)

log₅(125) = ln(125) / ln(5) = 4.8283 / 1.6094 = 3

Example — Antilog of 3

antilog(3) = 10³ = 1,000

How the Logarithm Calculator Works

Formula, assumptions, and calculation steps for this math tool.

Methodology

Math calculators apply the relevant arithmetic, algebraic, geometric, or numeric rule to the values entered and simplify the result where possible.

Calculation Steps

Read the values and operation selected.
Normalize signs, decimals, fractions, or units if needed.
Apply the mathematical rule or formula.
Format the answer and any intermediate values for checking.

Assumptions and Limits

Inputs must be within the supported domain of the operation.
Decimal answers may be rounded for readability.
Symbolic simplification is limited to the calculator scope.

Frequently Asked Questions

What is a logarithm?

A logarithm answers the question: to what exponent must we raise the base to get a given number? If logₐ(x) = y, then aʸ = x. Logarithms are the inverse of exponentiation.

What is the difference between log and ln?

log (common logarithm) uses base 10, while ln (natural logarithm) uses base e ≈ 2.71828. ln appears naturally in calculus, growth/decay models, and statistics, while log base 10 is common in chemistry (pH) and engineering (decibels).

What is the change-of-base formula?

The change-of-base formula allows computing logₐ(x) using any other base: logₐ(x) = log(x)/log(a) = ln(x)/ln(a). This is how calculators compute logarithms in arbitrary bases.

What is an antilogarithm?

The antilogarithm is the inverse of the logarithm. If log(x) = y, then the antilog of y is x = 10ʸ. For natural logs, the antilog of y is eʸ. Antilogs are used to reverse logarithmic transformations.

Real-World Applications

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Decibels (Sound Intensity)

Sound levels in dB = 10 × log₁₀(I/I₀), where I₀ is the threshold of hearing. A 10 dB increase represents a 10× increase in intensity — the logarithmic scale reflects human auditory perception.

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pH in Chemistry

pH = −log₁₀[H⁺] — a solution with H⁺ concentration of 10⁻⁷ mol/L has pH 7 (neutral). Each unit change in pH represents a 10× change in hydrogen ion concentration.

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Richter Scale (Earthquakes)

Earthquake magnitude M = log₁₀(A/A₀). Each whole number increase represents a 10× increase in wave amplitude and ~31.6× more energy — explaining why a magnitude 7 is vastly more destructive than a magnitude 5.

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Compound Interest & Time to Double

The time to double an investment at rate r: t = ln(2)/r (continuous compounding) or t = log(2)/log(1+r) (periodic). Logarithms invert the compound interest formula to solve for time.

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Algorithm Complexity

Binary search and efficient sort algorithms have time complexity O(log₂ n) — the number of steps grows as the binary logarithm of the input size, making them dramatically faster than linear algorithms for large datasets.

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Signal Strength & dBm

WiFi and cellular signal strength are measured in dBm (decibels relative to 1 milliwatt) — a logarithmic scale. A signal of −70 dBm is 10× weaker in power than −60 dBm.

Common Mistakes

Confusing log (base 10) with ln (base e)

"log" in mathematics often means natural log (base e); in engineering and everyday usage it usually means log base 10. Always check the convention — scientific calculators label them separately as "log" (base 10) and "ln" (natural).

Applying logarithm laws to sums

log(a + b) ≠ log(a) + log(b). The product rule states log(a × b) = log(a) + log(b). Applying the product rule to a sum is one of the most common logarithm errors in algebra.

Taking the logarithm of a negative number

Logarithms are only defined for positive real arguments. log(−5) is undefined in real number arithmetic (it exists in complex analysis as a complex number, but this is not relevant to most calculations).

Forgetting to apply change of base correctly

log_b(x) = log(x)/log(b) = ln(x)/ln(b). The same base must be used in both numerator and denominator. A common error is to use log in the numerator and ln in the denominator — producing a wrong result.

Confusing log(x²) with (log x)²

log(x²) = 2 log(x) by the power rule. But (log x)² means (log x) multiplied by itself — a completely different expression. The power rule applies inside the argument, not to the output of the logarithm.

Logarithm Laws Quick Reference

Law	Formula	Example
Product rule	log(ab) = log a + log b	log(100) = log(10) + log(10) = 2
Quotient rule	log(a/b) = log a − log b	log(1000/10) = 3 − 1 = 2
Power rule	log(aⁿ) = n·log a	log(10⁶) = 6·log(10) = 6
Change of base	log_b(x) = ln x / ln b	log₂(8) = ln 8 / ln 2 = 3
Identity	log_b(b) = 1	log₁₀(10) = 1; ln(e) = 1
Zero	log_b(1) = 0	log(1) = 0; ln(1) = 0

References

Napier, John. Mirifici Logarithmorum Canonis Descriptio. Edinburgh, 1614.
Stewart, James. Calculus: Early Transcendentals. Cengage, 2015.
Abramowitz, M. and Stegun, I.A. Handbook of Mathematical Functions. Dover, 1965.
NIST. NIST Digital Library of Mathematical Functions — Logarithm Function. dlmf.nist.gov, 2024.
Knuth, Donald. The Art of Computer Programming, Vol. 2 — Seminumerical Algorithms. Addison-Wesley, 1997.