Logarithm Calculator

Calculate logarithms with any base including natural log (ln), log base 10, and log base 2.

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1. Enter the value you want to find the logarithm of. 2. Select the base: common log (base 10), natural log (ln), binary log (base 2), or enter a custom base. 3. View the logarithm result along with the equivalent exponential form. 4. Review the change-of-base formula for understanding the conversion. 5. Click the copy button to copy the logarithm value.

About This Tool

The Logarithm Calculator computes logarithms for any base. It supports the most common logarithmic functions - log base 10 (common logarithm), ln (natural logarithm using base e), and log base 2 (binary logarithm) - as well as custom bases of your choosing. Enter a number and select or specify your base to get the result instantly.

Logarithms are the inverse of exponentiation and are essential across many fields. Scientists use logarithmic scales for earthquake magnitude (Richter scale), sound intensity (decibels), and acidity (pH). Engineers and computer scientists use log base 2 for algorithm complexity analysis. Mathematicians and physicists use natural logarithms extensively in calculus, differential equations, and growth models.

The tool provides high-precision results and includes helpful context like the relationship between the logarithm and its equivalent exponential expression. It also supports the change of base formula, showing you how to convert between different logarithmic bases. This makes it valuable both for quick calculations and for deepening your understanding of logarithmic relationships.

Formula / How It Works

log_b(x) = y means b^y = x | Change of base: log_b(x) = ln(x) / ln(b) | ln(e) = 1 | log(10) = 1

Frequently Asked Questions

A logarithm answers the question: "To what power must the base be raised to get this number?" For example, log base 2 of 8 = 3 because 2^3 = 8. In notation, log_b(x) = y means b^y = x.
log (common logarithm) uses base 10, ln (natural logarithm) uses base e (approximately 2.71828), and log2 (binary logarithm) uses base 2. They all follow the same principles but with different bases.
The change of base formula lets you calculate a logarithm of any base using a different base: log_b(x) = log_a(x) / log_a(b). For example, log_3(27) = ln(27) / ln(3) = 3.2958 / 1.0986 = 3.
The natural logarithm uses the mathematical constant e as its base, which appears naturally in calculus, compound interest, population growth, and many physical laws. The derivative of ln(x) is simply 1/x, making it fundamental in calculus.
In real numbers, no - logarithms are only defined for positive numbers. The logarithm of zero is undefined, and the logarithm of a negative number requires complex numbers (involving imaginary components).

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