The exponent that gets you back.
Enter a value and a base to see the logarithm, plus the natural, common and binary logs side by side. We even check the answer by raising the base back to the power.
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What this means
Logarithm across values
| Value x | logb(x) | ln(x) | log₁₀(x) |
|---|
How logarithms work
What Are Logarithms?
A logarithm answers the question: "To what power must the base be raised to get this number?" If 10² = 100, then log₁₀(100) = 2. Logarithms are the inverse of exponentiation. They transform multiplication into addition and are essential tools in mathematics, science, and engineering.
Types of Logarithms
Common log (log₁₀): used in pH scales, decibels, and earthquake magnitude. Natural log (ln, base e): used in calculus, continuous growth, and physics. Binary log (log₂): used in computer science and information theory. Each base serves specific applications in different fields.
Logarithm Properties
Key rules: log(a×b) = log(a) + log(b), log(a/b) = log(a) - log(b), log(a^n) = n×log(a), log(1) = 0, log_b(b) = 1. The change of base formula: log_b(x) = log(x) / log(b), which lets you convert between any bases.
Logarithms in Real Life
The Richter scale uses log₁₀ (each step = 10× energy). Decibels use log₁₀ for sound intensity. pH is -log₁₀ of hydrogen ion concentration. In finance, logarithmic returns are used for portfolio analysis. Computer scientists use log₂ to analyze algorithm complexity (O(log n)).
Common questions
Can you take the logarithm of zero or a negative number?
No. Logarithms are only defined for positive numbers (in real number mathematics). log(0) approaches negative infinity, and log of a negative number requires complex numbers.
What is the natural logarithm?
The natural logarithm (ln) uses base e ≈ 2.71828. It appears naturally in calculus and models continuous growth and decay. The derivative of ln(x) is 1/x, making it fundamental in differentiation and integration.
How do I convert between log bases?
Use the change of base formula: log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b). Most calculators only have ln and log₁₀ buttons, so this formula is essential for computing other bases.
What is e and why is it important?
Euler's number e ≈ 2.71828 is the base of the natural logarithm. It arises from the limit of (1 + 1/n)^n as n approaches infinity. It describes continuous compound growth and is one of the most important constants in mathematics.
Logarithms are defined only for positive values. Results use standard floating-point math, accurate to about 15 significant digits.