Logarithm Calculator
Compute the logarithm of any positive number in any base instantly. Use it for solving exponential equations, working with decibels, pH, or Richter scale values.
About this calculator
A logarithm answers the question: 'To what power must the base be raised to produce a given number?' Formally, log_b(x) = y means b^y = x. This calculator uses the change-of-base formula to handle any base: log_b(x) = ln(x) / ln(b), where ln denotes the natural logarithm. Common special cases include base 10 (the common logarithm, written log or log₁₀) and base e ≈ 2.71828 (the natural logarithm, written ln). Logarithms are essential in science and engineering — sound intensity is measured in decibels using log₁₀, earthquake magnitude uses the Richter scale (log₁₀), pH is −log₁₀[H⁺], and compound interest and population growth involve natural logarithms. Inputs must satisfy x > 0 and b > 0 with b ≠ 1.
How to use
Find log₂(32) — the power to which 2 must be raised to equal 32. Enter Number = 32 and Base = 2. The calculator applies the change-of-base formula: log₂(32) = ln(32) / ln(2) = 3.4657… / 0.6931… = 5. The result is 5, which you can verify: 2⁵ = 32. Now try a non-integer result: log₁₀(50) = ln(50) / ln(10) ≈ 3.912 / 2.303 ≈ 1.699, meaning 10^1.699 ≈ 50.
Frequently asked questions
What is the difference between log, ln, and log base 2?
These are all logarithms but with different bases. 'log' without a specified base conventionally means log base 10 (the common logarithm), widely used in engineering and the sciences. 'ln' is the natural logarithm with base e ≈ 2.71828, fundamental in calculus, continuous growth models, and physics. Log base 2 (log₂, or the binary logarithm) appears in computer science — for example, the number of bits needed to represent n values is ⌈log₂(n)⌉. All three are interconvertible using the change-of-base formula.
Why can you not take the logarithm of zero or a negative number?
The logarithm log_b(x) asks what power of b equals x. Since any positive base b raised to any real power always produces a strictly positive result, there is no real exponent that yields zero or a negative number. Mathematically, the domain of any real logarithm function is restricted to x > 0. Attempting log(0) approaches negative infinity as a limit, while log of a negative number requires complex number theory (involving imaginary components) and is outside the scope of real-valued logarithm calculators.
How are logarithms used in real-world measurements like decibels and pH?
Logarithmic scales compress enormous ranges of values into manageable numbers. Sound intensity in decibels is calculated as dB = 10 × log₁₀(I / I₀), where I₀ is the threshold of human hearing — a 10 dB increase represents a 10× increase in intensity. The pH scale measures hydrogen ion concentration as pH = −log₁₀[H⁺], so each pH unit represents a 10-fold change in acidity. The Richter scale similarly uses log₁₀ so that a magnitude 7 earthquake releases about 31.6 times more energy than a magnitude 6. These applications show why understanding logarithms is practically useful.