Logarithm Calculator

Calculate logarithms with any base including natural log (ln), common log (log₁₀), and custom bases

Logarithm Calculator

Calculation Type

Base (b > 0, b ≠ 1)

Argument (x > 0)

Logarithm Reference

Basic Definition

log₍ᵦ₎(x) = y ⟺ b^y = x

"b to what power equals x?"

Common Logarithms

ln(x)Natural log (base e)

log₁₀(x)Common log (base 10)

log₂(x)Binary log (base 2)

Logarithm Properties

log₍ᵦ₎(xy) = log₍ᵦ₎(x) + log₍ᵦ₎(y)

log₍ᵦ₎(x/y) = log₍ᵦ₎(x) - log₍ᵦ₎(y)

log₍ᵦ₎(x^n) = n·log₍ᵦ₎(x)

log₍ᵦ₎(b) = 1

log₍ᵦ₎(1) = 0

Change of Base Formula

log₍ₐ₎(x) =

log₍ᵦ₎(x)

log₍ᵦ₎(a)

Special Values

ln(e) = 1

log₁₀(10) = 1

log₂(2) = 1

ln(1) = 0

log₁₀(100) = 2

log₂(8) = 3

Frequently Asked Questions

A logarithm is the inverse operation of exponentiation. log₍ᵦ₎(x) = y means that b^y = x. For example, log₍₂₎(8) = 3 because 2³ = 8. Logarithms help solve exponential equations and are used in many scientific applications.

Tips

Enter the dimensions in the same unit for consistency.
Results update automatically as you type.
Use the unit selector to convert between different measurement systems.

Logarithm Calculator