Exponent & Log Calculator

This Exponent & Log Calculator allows you to compute both exponents (aⁿ) and logarithms (logₐ(b)) instantly. It supports large numbers, negative exponents, fractional exponents, natural logs, base-10 logs, and custom logarithm bases.

What Are Exponents?

Exponents are a shorthand way to express repeated multiplication. For example, 2⁵ means multiplying 2 by itself 5 times: 

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

Exponent Rules

  • Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ
  • Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
  • Power rule: (aᵐ)ⁿ = aᵐⁿ
  • Zero exponent: a⁰ = 1
  • Negative exponent: a⁻ⁿ = 1 / aⁿ
  • Fractional exponent: a^(1/n) = ⁿ√a

Examples

  • 3⁴ = 81
  • 5⁻² = 1/25
  • 9^(1/2) = 3
  • 2¹⁰ = 1024

What Are Logarithms?

A logarithm tells you the power needed to reach a certain number. For example: 

log₂(32) = 5 because 2⁵ = 32

Common Types of Logarithms

  • log₁₀(x) — common logarithm
  • ln(x) — natural logarithm (base e)
  • logₐ(b) — logarithm with any custom base

Change of Base Formula

logₐ(b) = ln(b) / ln(a)

Examples

  • log₁₀(1000) = 3
  • ln(e²) = 2
  • log₂(1/8) = -3

Real-World Applications

  • Earthquake magnitude (Richter scale)
  • Sound intensity (decibels)
  • pH in chemistry
  • Compound interest & finance
  • Radioactive decay
  • Population growth
  • Computer science (log₂ operations)

Frequently Asked Questions (FAQ)

Because 1 raised to any power always equals 1, so log₁(b) is undefined.
Yes. log(a) is negative when 0 < a < 1.
Because no exponent of a positive number can produce 0.
ln(x) is logarithm base e (≈2.71828), fundamental in natural growth and decay.
Yes. Fractional exponents represent roots: a^(1/n) = ⁿ√a.
A compact way to write very large or very small numbers using powers of 10.
Because binary operations scale with powers of 2.
Yes. A negative exponent means 1 divided by the positive exponent value.
Yes. logₐ(b) answers “what exponent of a gives b?”.
Yes, this is one of the standard logarithm identities.
Yes, results are converted to scientific notation automatically.
Yes. ln(x) simply means log base e.
Logarithms of negative numbers are not defined in the real number system.
Because it keeps the exponent rules consistent across integers.
log usually means base 10; ln means base e.