Logarithm Calculator

How to Use the Logarithm Calculator

Enter a positive number in the value field and choose your base. Type e for the natural logarithm, 10 for the common logarithm, or any positive number other than 1 for a custom base. The calculator displays the result for your chosen base alongside ln, log₁₀, and log₂ for easy comparison.

Logarithms are the inverse of exponentiation and appear throughout science, engineering, and finance. Understanding them unlocks concepts from pH scales and decibels to compound interest and algorithm complexity. This tool computes all common logarithmic forms simultaneously so you can see the relationships at a glance.

Understanding Logarithms

If b^x = y, then log_b(y) = x. The base b must be positive and not equal to 1. The natural logarithm (ln) uses base e ≈ 2.71828, which arises naturally in calculus and exponential growth. The common logarithm (log₁₀) is used in orders of magnitude, decibels, and the Richter scale.

The Change of Base Formula

You can convert between any two bases using log_b(x) = ln(x) / ln(b). This is exactly how the calculator computes custom-base logarithms. For instance, log₅(125) = ln(125) / ln(5) = 3, because 5³ = 125.

Key Logarithm Properties

Logarithms follow three fundamental rules: log(xy) = log(x) + log(y), log(x/y) = log(x) − log(y), and log(xⁿ) = n · log(x). These properties simplify multiplication into addition, which is why logarithms were historically used in slide rules and log tables for computation.

Applications of Logarithms

Logarithms measure earthquake intensity on the Richter scale (each whole number is a tenfold increase), sound levels in decibels, and acidity on the pH scale. In finance, they model compound interest and continuous growth. In computer science, logarithmic time complexity describes efficient algorithms like binary search. Information theory uses log₂ to measure entropy in bits.

Frequently Asked Questions

What is a logarithm?

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

What is the difference between ln, log, and log2?

ln uses base e (~2.718), log typically means base 10, and log2 uses base 2. They are all related by the change-of-base formula.

Why can't you take the logarithm of a negative number?

In real mathematics, logarithms are only defined for positive numbers because no real exponent of a positive base can produce a negative result.

What is the change of base formula?

log_b(x) = log_a(x) / log_a(b). This lets you convert any logarithm to a more convenient base.

Where are logarithms used in real life?

Earthquake measurement, sound levels, pH scales, radioactive decay, compound interest, information theory, and algorithm analysis all rely on logarithms.