Logarithm Calculator
Calculate logarithms in any base or find the antilog. Supports log base 10, natural log (ln), log base 2, and custom bases.
What Is a Logarithm?
A logarithm is the inverse of exponentiation. It answers the question: what power do I raise the base to in order to get a certain number? If 10³ = 1,000, then log₁₀(1,000) = 3. The base tells you which number is being raised to a power, and the result is the exponent.
logᵢ(x) = y means bʸ = x
Change of base: logᵢ(x) = ln(x) / ln(b)
Common Logarithm Types
Common log (log₁₀) uses base 10 and is written simply as "log." It is used in decibel scales, pH calculations, and earthquake magnitudes (Richter scale). Natural log (ln) uses base e (2.71828...) and appears throughout calculus, physics, and continuous growth formulas. Binary log (log₂) uses base 2 and is essential in computer science for algorithm complexity analysis.
Logarithm Rules
There are three key rules that make logarithms powerful for simplifying calculations. The product rule: log(a x b) = log(a) + log(b). The quotient rule: log(a / b) = log(a) - log(b). The power rule: log(aⁿ) = n x log(a). These rules turn multiplication into addition, division into subtraction, and exponents into multiplication.
What is the natural number e?
e is approximately 2.71828 and is one of the most important constants in mathematics. It is the base of the natural logarithm and shows up in compound interest, population growth, radioactive decay, and probability theory. Like pi, it is irrational and goes on forever without repeating.
Can you take the log of a negative number?
Not with real numbers. The log of a negative number is undefined in the real number system because no real power of a positive base gives a negative result. In complex analysis, you can define it using imaginary numbers, but for standard math classes, log(x) requires x to be greater than 0.
What is an antilog?
The antilog (or antilogarithm) reverses the logarithm. If log₁₀(1000) = 3, then antilog₁₀(3) = 1000. Mathematically, antilog is just exponentiation: antilog of x in base b equals b raised to the power x.
Why is log(1) always 0?
Because any number raised to the power 0 equals 1. So log in any base of 1 is always 0. For example, 10⁰ = 1, e⁰ = 1, 2⁰ = 1, and so on.