Logarithm Calculator

A logarithm answers “what power do I raise the base to, to get this number?” Pick a base, enter a positive number, and get the log with a check.

Logarithm

log₁₀(1,000)3
Natural log ln(x)
6.907755279
Binary log log₂(x)
9.9657842847

Working

  1. log₁₀(1,000) asks: 10 to what power equals 1,000?
  2. log₁₀(1,000) = 3
  3. Check: 10^3 = 1,000

Antilogarithm (base to a power)

10^2100

Working

  1. The antilogarithm undoes the log: if log base 10 of x = 2, then x = 10^2 = 100

Definition

log_b(x) = y ⇔ b^y = x

b
the base (positive, not 1)
x
the number (must be positive)
y
the logarithm

So log₁₀(1,000) = 3 because 10³ = 1,000, and log₂(32) = 5 because 2⁵ = 32. Logarithms of numbers between 0 and 1 are negative: log₁₀(0.01) = −2.

Change of base

log_b(x) = ln(x) ÷ ln(b)

log₃(20)

  1. ln(20) ≈ 2.995732
  2. ln(3) ≈ 1.098612
  3. 2.995732 ÷ 1.098612 ≈ 2.726833

log₃(20) ≈ 2.7268, and 3^2.7268 ≈ 20.

Which base to use

  • Base 10 (log): decibels, the Richter and pH scales, orders of magnitude.
  • Base e (ln, e ≈ 2.71828): continuous growth and decay, calculus, statistics.
  • Base 2 (log₂): computing — how many bits are needed, how many times a set can be halved.

Where logarithms are undefined

The logarithm of zero or a negative number has no real value, because no real power of a positive base gives zero or a negative number. The base must also be positive and not 1 — every power of 1 is 1. The calculator reports these cases instead of returning a misleading number.