# Logarithm Calculator

Calculate logarithms with any base: common log (base 10), natural log (ln), binary log (base 2), or custom bases. Free online logarithm calculator.

## What this calculates

Calculate the logarithm of any positive number with any base using this free online tool. Choose from common logarithm (base 10), natural logarithm (base e), binary logarithm (base 2), or enter your own custom base.

## Inputs

- **Logarithm Base** — options: Common Log (base 10), Natural Log (base e), Binary Log (base 2), Custom Base — Select the base for the logarithm calculation.
- **Custom Base** — Only used when 'Custom Base' is selected above.
- **Value (x)** — The number to find the logarithm of. Must be positive.

## Outputs

- **Logarithm Result** — The logarithm of the value with the specified base.
- **Expression** — formatted as text — The logarithmic expression evaluated.
- **Antilog (base^result = value)** — Verification: the base raised to the result should equal the input value.

## Details

A logarithm answers the question: 'To what power must the base be raised to produce a given number?' If b^y = x, then log_b(x) = y.

Common Types of Logarithms

- Common Logarithm (log₁₀): Base 10. Used in science, engineering, and the decibel scale. For example, log₁₀(1000) = 3 because 10^3 = 1000.

- Natural Logarithm (ln): Base e (approximately 2.71828). Essential in calculus, physics, and continuous growth/decay models. For example, ln(e^2) = 2.

- Binary Logarithm (log₂): Base 2. Fundamental in computer science, information theory, and algorithm analysis. For example, log₂(256) = 8.

Key Logarithm Properties

- log_b(m x n) = log_b(m) + log_b(n) (Product Rule)

- log_b(m / n) = log_b(m) - log_b(n) (Quotient Rule)

- log_b(m^k) = k x log_b(m) (Power Rule)

- log_b(1) = 0 for any valid base

- log_b(b) = 1 for any valid base

- Change of base: log_b(x) = log_c(x) / log_c(b)

Logarithms are the inverse of exponentiation and are indispensable in fields ranging from seismology (Richter scale) to music (frequency intervals) to finance (continuously compounded interest).

## Frequently Asked Questions

**Q: What is a logarithm?**

A: A logarithm is the inverse of exponentiation. log_b(x) = y means that b raised to the power y equals x. For example, log₁₀(100) = 2 because 10^2 = 100. Logarithms tell you what exponent is needed to reach a given value.

**Q: What is the difference between common log and natural log?**

A: The common logarithm (log or log₁₀) uses base 10 and is used in applied sciences and engineering. The natural logarithm (ln) uses base e (approximately 2.71828) and is fundamental in pure mathematics, calculus, and models involving continuous growth or decay.

**Q: Why can you not take the logarithm of zero or a negative number?**

A: In the real number system, there is no exponent that makes a positive base equal zero or a negative number. For example, no value of y satisfies 10^y = 0 or 10^y = -5. Logarithms of negative numbers do exist in the complex number system but are outside the scope of this calculator.

**Q: What is the change of base formula?**

A: The change of base formula allows you to compute a logarithm in any base using a different base: log_b(x) = log_c(x) / log_c(b). This is useful because most calculators only have buttons for base 10 and base e. For example, log_5(125) = ln(125) / ln(5) = 4.828 / 1.609 = 3.

**Q: Where are logarithms used in real life?**

A: Logarithms appear in many real-world applications: the Richter scale for earthquake magnitude, decibels for sound intensity, pH for acidity, information entropy in computer science, compound interest in finance, and radioactive decay half-lives in physics. They are essential whenever quantities span many orders of magnitude.

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Source: https://vastcalc.com/calculators/math/logarithm
Category: Math
Last updated: 2026-04-21