Logarithm Calculator
Calculate logarithms with any base. Supports natural log (ln), log base 10, and custom bases.
What Is the Logarithm Calculator?
The Logarithm Calculator computes logarithms in any base — base 10 (common log), base e (natural log, ln), base 2 (binary log), or any custom base you specify. You can compute all three standard logarithms at once, or focus on a specific base.
Formula
How to Use
Enter the value x (must be positive). Select the base: "All" computes log₁₀, ln, and log₂ simultaneously; specific bases compute only that logarithm; "Custom" lets you enter any base. Click Calculate to see the results with the change-of-base verification.
Example Calculation
log₁₀(1000) = 3 (since 10³ = 1000) ln(e²) = 2 (since e² = e²) log₂(64) = 6 (since 2⁶ = 64) Custom: log₅(125): ln(125)/ln(5) = 4.8283/1.6094 = 3 (since 5³ = 125 ✓) log₁₀(500): = log₁₀(5×100) = log₁₀(5) + 2 ≈ 0.6990 + 2 = 2.699
Understanding Logarithm
Logarithms were invented by John Napier in 1614 to simplify multiplication and division of large numbers. Before calculators, scientists and engineers used log tables to convert multiplication into addition: log(a×b) = log(a) + log(b).
The natural logarithm ln is ubiquitous in mathematics and science because it is the natural inverse of the exponential function e^x. It appears in compound interest, population growth, radioactive decay, entropy, and the famous equation P = P₀e^(rt).
pH in chemistry (pH = −log₁₀[H⁺]), the Richter scale for earthquakes, and decibels for sound intensity are all logarithmic scales. Using logarithms lets us work with quantities spanning many orders of magnitude in a compact way.
Frequently Asked Questions
What is a logarithm?
A logarithm answers the question: "To what power must the base be raised to get x?" logb(x) = y means b^y = x. Logarithms are the inverse of exponentials.
What is the natural logarithm?
The natural logarithm ln(x) uses base e ≈ 2.71828 (Euler's number). It appears naturally in calculus because d/dx[ln(x)] = 1/x, making it the simplest antiderivative of 1/x.
Why is log₂ important?
Log base 2 is the language of computer science. It measures the number of bits needed to represent n possibilities: log₂(1024) = 10, so 10 bits can represent 1024 values.
Can I take the log of a negative number?
No. The logarithm is only defined for positive real numbers (x > 0). The base must also be positive and not equal to 1. Logarithms of negative numbers exist in complex number theory.
What is the change of base formula?
The change of base formula logb(x) = ln(x)/ln(b) allows computation of any logarithm using any other logarithm. Most calculators only provide log₁₀ and ln, so this formula unlocks all other bases.
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