What is a logarithm and how does it relate to exponents?

A logarithm is the inverse operation of exponentiation. If b^y = x, then log_b(x) = y. In other words, the logarithm answers 'what power do I raise the base to, to get this number?' For example, log₂(8) = 3 because 2³ = 8.

What are the most common logarithm bases?

The most common bases are: base 10 (common logarithm, written as log), base e ≈ 2.718 (natural logarithm, written as ln), and base 2 (binary logarithm, used in computer science). Each has specific applications - base 10 for general calculations, ln for calculus and growth/decay, and log₂ for information theory.

What are the key properties of logarithms?

Important logarithm properties include: log(xy) = log(x) + log(y), log(x/y) = log(x) - log(y), log(x^n) = n·log(x), and log_b(b) = 1. These properties allow you to simplify complex logarithmic expressions and solve logarithmic equations.

How do I change the base of a logarithm?

Use the change of base formula: log_b(x) = log_a(x) / log_a(b), where a is any convenient base (usually 10 or e). For example, to calculate log₂(100) using common logarithms: log₂(100) = log₁₀(100) / log₁₀(2) ≈ 6.644.

Where are logarithms used in real-world applications?

Logarithms are everywhere! They measure earthquake intensity (Richter scale), sound intensity (decibels), acidity (pH scale), and stellar brightness. They're essential in compound interest calculations, population growth models, algorithm complexity analysis (Big O notation), and signal processing.

Math

Logarithm Calculator

Calculate logarithms with any base.

How to Use This Calculator

Select the base

Choose the logarithm base you want to use. Common options include base 10 (common log), base e (natural log), base 2 (binary log), or enter a custom base for specialized calculations.

Enter the argument

Input the number you want to find the logarithm of (must be positive). This is the value x in log_b(x).

Interpret the result

The calculator displays the logarithm value, which represents the power to which the base must be raised to produce your input number. The result may also show verification (base^result = input) to help confirm understanding.