Logarithm Calculator

How it works

A logarithm answers the question "what power of the base gives this number?". log10(1000) = 3 because 10^3 = 1000; log2(1024) = 10 because 2^10 = 1024; ln(x) is the logarithm to base e ≈ 2.71828. The antilog is the reverse: given the exponent y, it returns b^y.

Calculators and tables only provide a few bases, usually 10 and e. Any other base comes from the change-of-base formula: log_b(x) = ln(x) ÷ ln(b). The Working line shows that division so you can reproduce the answer on a basic scientific calculator, and the Check line raises the base back to the result to confirm it.

Logarithms turn multiplication into addition, which is why they underpin decibels, pH, the Richter scale, and the doubling-time question in finance: the number of periods for an investment to reach a target is log(target ÷ start) ÷ log(1 + rate).

Formula

log_b(x) = y   ⇔   b^y = x          (x > 0, b > 0, b ≠ 1)
log_b(x) = ln(x) ÷ ln(b)            (change of base)
antilog:  b^y

Example

log10(1000): ln(1000) ÷ ln(10) = 6.907755 ÷ 2.302585 = 3, because 10^3 = 1000.

log2(10) = 3.321928, which is how many bits it takes to hold one decimal digit. With base 5, log_5(125) = 3.

Antilog with base 10 and exponent 2.5: 10^2.5 = 316.227766.

Assumptions and limitations

• The logarithm is only defined for numbers greater than zero; zero and negative numbers are rejected rather than reported as undefined or complex.
• The base must be positive and not 1. Bases between 0 and 1 are allowed and give negative logarithms for numbers above 1.
• Results are cleaned to 15 significant digits so that exact cases such as log10(1000) = 3 come out exact, and are shown to 6 decimal places.

Frequently asked questions