Logarithm & Exponent Calculator

Calculate logarithms, natural log, antilog, and exponents with any base

Select Operation

Choose the type of calculation

Input Values

Enter values for logarithm calculation

The number to find the logarithm of (must be positive)

Common bases: 10 (common log), 2 (binary), e ≈ 2.718 (natural)

Common Logarithms

log₁₀(10) = 1
Common logarithm of 10
log₁₀(100) = 2
Common logarithm of 100
log₁₀(1000) = 3
Common logarithm of 1000
ln(e) = 1
Natural log of e ≈ 2.718
log₂(8) = 3
Binary logarithm of 8
log₂(1024) = 10
Binary logarithm of 1024

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About Logarithm & Exponent Calculator

Calculate logarithms with any base, natural logarithms (ln), antilogs, and exponents with our comprehensive calculator. Perfect for students, engineers, scientists, and anyone working with exponential and logarithmic functions.

What is a Logarithm?

A logarithm answers the question: "To what power must we raise the base to get this number?" If b^y = x, then log₍ᵦ₎(x) = y. Logarithms are the inverse operation of exponentiation and are fundamental in mathematics, science, and engineering.

Types of Logarithms

  • Common Logarithm (log₁₀): Base 10 logarithm, widely used in science and engineering
  • Natural Logarithm (ln): Base e ≈ 2.71828, essential in calculus and exponential growth/decay
  • Binary Logarithm (log₂): Base 2, commonly used in computer science and information theory
  • Custom Base: Any positive base (except 1) can be used for specialized calculations

Understanding Antilog

Antilog is the inverse operation of logarithm. If log₍ᵦ₎(x) = y, then antilog₍ᵦ₎(y) = x. In other words, antilog calculates b^y. This is useful for reversing logarithmic transformations and solving exponential equations.

Exponents Explained

An exponent represents repeated multiplication. a^n means multiply 'a' by itself 'n' times. For example, 2^3 = 2 × 2 × 2 = 8. Exponents can be positive, negative, or fractional:

  • Positive exponents: Standard multiplication (2^3 = 8)
  • Negative exponents: Reciprocals (2^-3 = 1/8 = 0.125)
  • Fractional exponents: Roots (4^(1/2) = √4 = 2)
  • Zero exponent: Any number to the power of 0 equals 1 (5^0 = 1)

Logarithm Properties

  • Product Rule: log(xy) = log(x) + log(y)
  • Quotient Rule: log(x/y) = log(x) - log(y)
  • Power Rule: log(x^n) = n × log(x)
  • Base Identity: log₍ᵦ₎(b) = 1
  • Zero Identity: log₍ᵦ₎(1) = 0
  • Change of Base: log₍ᵦ₎(x) = log(x) / log(b)

Applications of Logarithms

  • Science: pH scale, Richter scale, decibel scale
  • Finance: Compound interest, investment growth calculations
  • Biology: Population growth, radioactive decay
  • Computer Science: Algorithm complexity, data compression
  • Engineering: Signal processing, control systems
  • Statistics: Log-normal distributions, data transformation

Common Logarithm Values

  • log₁₀(10) = 1
  • log₁₀(100) = 2
  • log₁₀(1000) = 3
  • ln(e) = 1 (where e ≈ 2.71828)
  • log₂(8) = 3
  • log₂(1024) = 10

Tips for Using the Calculator

  • For logarithms, the value must be positive (x > 0)
  • The base must be positive and not equal to 1 (b > 0, b ≠ 1)
  • Use natural log (ln) for calculations involving e
  • Antilog is equivalent to raising the base to the given power
  • Results are shown in both exponential and decimal notation for precision

Important Notes

Logarithms of negative numbers and zero are undefined in real numbers. The base of a logarithm cannot be 1 because 1 raised to any power is always 1. For very large or very small results, scientific notation (exponential form) is used to maintain precision.