Logarithm Calculator

Use this logarithm calculator to compute common logarithm (base 10), natural logarithm (base e), or logarithm with any custom base. Get step-by-step solutions instantly.

Logarithm Calculator

Select the type of logarithm, enter a positive number, and click Calculate to find the logarithm value.

What is a Logarithm?

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

For example, since 10³ = 1000, the logarithm of 1000 to base 10 (the common logarithm) is 3, or log₁₀(1000) = 3. Similarly, since e² ≈ 7.389, the natural logarithm of 7.389 is approximately 2, or ln(7.389) ≈ 2.

Logarithm Formulas

Here are the key formulas and properties of logarithms:

$\log_b(x) = y \iff b^y = x$

The common logarithm (base 10) is often written without the base:

$\log(x) = \log_{10}(x)$

The natural logarithm (base e) is denoted by ln:

$\ln(x) = \log_e(x)$

The change of base formula allows us to compute logarithms with any base:

$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$

Applications of Logarithms

Science

Used in pH measurements, earthquake magnitude (Richter scale), sound intensity (decibels), and radioactive decay calculations.

Engineering

Applied in signal processing, control systems, and information theory for measuring signal strength and data compression.

Finance

Used in compound interest calculations, modeling exponential growth, and analyzing returns on investments over time.

Computer Science

Essential in algorithm analysis (Big O notation), data compression, machine learning, and information theory.

How to Use This Calculator

  1. Select the type of logarithm you want to calculate (common, natural, or custom base).
  2. Enter the number for which you want to find the logarithm (must be positive).
  3. If you selected custom base, enter the base value (must be positive and not equal to 1).
  4. Click 'Calculate' to compute the logarithm and see the step-by-step solution.