Logarithm Calculator

Quick start

1. Type the number x, for example 64.
2. Type the base, for example 4.
3. Read log base 4 of 64, plus ln, log₁₀ and log₂ of the same number.

What a logarithm answers

The log of x in base b asks: what power of b gives x? log₄ 64 = 3, because 4³ = 64. log₁₀ 1000 = 3, because 10³ = 1000.

Change of base

Most calculators only have ln and log₁₀. To use any base, divide: log_b x = ln x ÷ ln b. For log₃ 50: ln 50 ÷ ln 3 = 3.912 ÷ 1.0986 ≈ 3.561.

Log rules

• log(xy) = log x + log y
• log(x ÷ y) = log x − log y
• log(xⁿ) = n · log x
• log_b 1 = 0 and log_b b = 1

Real-world logs

Logarithms measure things that grow by multiplying. Each step up the Richter scale is about ten times more ground movement, and the pH scale counts powers of ten in acidity. log₂ tells you how many times something doubles, such as how many bits are needed to count to a number.

Common questions