Logarithm Calculator
Calculate the logarithm of any positive number using any base. Enter the number and the base to find logbase(number).
Natural Logarithm Calculator (ln)
Calculate the natural logarithm (base e) of any positive number. ln(x) = loge(x), where e ≈ 2.71828.
Antilogarithm Calculator
Calculate the antilogarithm (inverse logarithm) of a value. If logb(x) = y, then antilog = by = x.
Logarithm Equation Solver
Solve logb(x) = y for any missing variable. Enter two of the three values (base, number, result) and solve for the third.
Common Logarithm Reference Table
| Number (x) | log₂(x) | log₁₀(x) | ln(x) |
|---|
What is a Logarithm?
A logarithm answers the question: “To what power must a base be raised to produce a given number?” If by = x, then logb(x) = y. Logarithms are the inverse operation of exponentiation.
log2(8) = 3 because 2³ = 8
loge(e²) = 2 because e² = e²
Types of Logarithms
The three most common logarithms are the common logarithm (base 10, written log), the natural logarithm (base e ≈ 2.71828, written ln), and the binary logarithm (base 2, written log₂). Any base can be used through the change-of-base formula.
Natural log: ln(x) = loge(x)
Binary log: log₂(x)
Change of base: logb(x) = ln(x) / ln(b) = log(x) / log(b)
Logarithm Rules
Logarithms follow a set of rules that simplify complex calculations. These rules allow multiplication to become addition, division to become subtraction, and exponentiation to become multiplication.
Quotient rule: logb(x / y) = logb(x) − logb(y)
Power rule: logb(xn) = n · logb(x)
Identity: logb(b) = 1
Zero rule: logb(1) = 0
Antilogarithm
The antilogarithm is the inverse of the logarithm. If logb(x) = y, then the antilog base b of y equals x. It is computed as by.
antilog2(4) = 2⁴ = 16
antiloge(1) = e¹ ≈ 2.71828
Real-World Applications
Logarithms are used across science, engineering, and everyday life. The Richter scale measures earthquake magnitude logarithmically. Decibels (dB) measure sound intensity using base-10 logarithms. pH values in chemistry, compound interest calculations, and information theory (bits of data) all rely on logarithms.
Decibels: dB = 10 · log10(P / P₀)
Compound interest: t = ln(A/P) / (n · ln(1 + r/n))