Logarithm Calculator

Calculate the logarithm of any number with any base. Get common log, natural log, and custom base results with steps.

Enter Values

Result

log10(100)
--
Common Log (log10) --
Natural Log (ln) --
Binary Log (log2) --
Antilog (bresult) --

Step-by-Step Solution

log_b(x) = ln(x) / ln(b)

Understanding Logarithms

A logarithm answers the question: "To what power must the base be raised to produce a given number?" In other words, if by = x, then logb(x) = y. Logarithms are the inverse of exponentiation and are fundamental to many areas of mathematics, science, and engineering.

Types of Logarithms

Common Logarithm (log10)

Base 10 logarithm, widely used in science, engineering, and the decibel scale.

log(x) = log10(x)

Natural Logarithm (ln)

Base e (2.71828...) logarithm, fundamental in calculus and exponential growth.

ln(x) = loge(x)

Binary Logarithm (log2)

Base 2 logarithm, essential in computer science and information theory.

log2(x) = ln(x) / ln(2)

Change of Base Formula

Convert between any two bases using the change of base formula.

logb(x) = ln(x) / ln(b)

Logarithm Properties

  • Product rule: logb(xy) = logb(x) + logb(y)
  • Quotient rule: logb(x/y) = logb(x) - logb(y)
  • Power rule: logb(xn) = n * logb(x)
  • Identity: logb(b) = 1 for any valid base
  • Zero exponent: logb(1) = 0 for any valid base

Applications of Logarithms

Logarithms are used extensively in many fields. In science, the pH scale, Richter scale, and decibel scale are all logarithmic. In finance, compound interest and growth calculations rely on natural logarithms. In computer science, logarithms describe the efficiency of algorithms like binary search (O(log n)). In information theory, the bit (binary digit) is defined using log base 2.

Important Notes

  • The argument (x) must be positive. Logarithms of zero or negative numbers are undefined in real numbers.
  • The base must be positive and not equal to 1.
  • logb(b) always equals 1, and logb(1) always equals 0.