Math & Algebra Tool

Logarithm Calculator

Calculate logs for any base with steps.

Σ The Formula

log_b(x) = y means b^y = x

Real World Examples

Common Log
log₁₀(1000) = 3 because 10³ = 1000
Natural Log
ln(e) = 1 because e¹ = e
Binary Log
log₂(8) = 3 because 2³ = 8
Fraction
log₁₀(0.1) = -1

# About This Calculator

A Logarithm answers the question: "To what power must we raise a base number to get a specific result?" It is the inverse operation of exponentiation. If 2³ = 8, then log₂(8) = 3.

Logarithms are everywhere in science and nature. The Richter scale (earthquakes), pH scale (acidity), and Decibel scale (sound) are all logarithmic. This means an increase of 1 on the scale represents a 10-fold increase in intensity.

There are two most common bases: Base 10 (Common Log, written as "log") and Base *e* (Natural Log, written as "ln", where e ≈ 2.718). Computer science frequently uses Base 2 (Binary Log).

This tool computes the logarithm for any positive number and any positive base, showing the exponential relationship clearly.

How To Use

  1. Enter the Number you want to find the log of.
  2. Enter the Base (default is 10).
  3. Click Calculate.

Frequently Asked Questions

What is a Natural Log (ln)?+

It is a logarithm with base 'e' (Euler's number, approx 2.718). It is standard in calculus and physics for calculating growth and decay.

Why can't I take the log of a negative number?+

Because there is no real power you can raise a positive base to that gives a negative result. Positive^anything is always positive.

What is log(1)?+

Log of 1 is always 0, regardless of the base. This is because any number raised to the power of 0 equals 1 (b^0 = 1).

How do I convert between logs and exponents?+

The equation y = log_b(x) is equivalent to x = b^y. Start with the base, go across the equal sign for the power, and equals the number inside.

What is the change of base formula?+

log_b(x) = log_k(x) / log_k(b). This lets you calculate any log using only the standard ln or log10 buttons on a calculator.

Is Logarithm Calculator free to use?+

Yes, Logarithm Calculator on Matheric is completely free to use. We believe in accessible education and utility for everyone.

About

A Logarithm answers the question: "To what power must we raise a base number to get a specific result?" It is the inverse operation of exponentiation. If 2³ = 8, then log₂(8) = 3.

Logarithms are everywhere in science and nature. The Richter scale (earthquakes), pH scale (acidity), and Decibel scale (sound) are all logarithmic. This means an increase of 1 on the scale represents a 10-fold increase in intensity.

There are two most common bases: Base 10 (Common Log, written as "log") and Base *e* (Natural Log, written as "ln", where e ≈ 2.718). Computer science frequently uses Base 2 (Binary Log).

This tool computes the logarithm for any positive number and any positive base, showing the exponential relationship clearly.

Related Tools

Fraction Calculator

Fraction Simplifier

Fraction to Decimal

Decimal to Fraction