A logarithm calculator is an essential mathematical tool that helps compute logarithms with any base, including base 10 (common logarithm) and base e (natural logarithm or ln). Our calculator provides accurate results for logarithms, antilogarithms, and can solve logarithmic equations with step-by-step explanations.

Number (x):

Base (b):

Logarithm Value:

Base (b):

Logarithmic Equation:

About Logarithms

The logarithm of a number is the exponent to which another fixed value (the base) must be raised to produce that number. Mathematically, if b^y = x, then log_b(x) = y.

Common Logarithm Types

Common Logarithm (log₁₀): Base 10, widely used in science and engineering
Natural Logarithm (ln): Base e (≈2.71828), used in advanced mathematics and physics
Binary Logarithm (log₂): Base 2, important in computer science

Logarithm Properties

Property	Formula
Product Rule	log_b(xy) = log_b(x) + log_b(y)
Quotient Rule	log_b(x/y) = log_b(x) - log_b(y)
Power Rule	log_b(x^y) = y·log_b(x)
Change of Base	log_b(x) = log_k(x) / log_k(b)

Practical Applications

Science: Measuring pH, sound intensity (decibels), and earthquake magnitude
Finance: Compound interest calculations and exponential growth models
Computer Science: Algorithm complexity analysis (Big O notation)
Engineering: Signal processing and control systems
Statistics: Logarithmic transformations for data normalization

Logarithm Calculator

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Logarithmic Function Graph