Log & Antilog Calculator

Calculate logarithms and antilogs for any base. Common log (base 10), natural log (base e) and custom base.

Choose a mode and enter values to calculate.

Understanding logarithms

A logarithm answers the question: "To what power must we raise the base to get this number?" log₁₀(1000) = 3 because 10³ = 1000.

Definition
logᵦ(x) = y ↔ bʸ = x
log₁₀(100) = 2 because 10² = 100. ln(e²) = 2 because e² = e².
Change of Base
logᵦ(x) = log(x) / log(b)
log₂(8) = log(8)/log(2) = 0.9031/0.3010 = 3. Because 2³ = 8.
Log Properties
log(ab) = log(a) + log(b)
log(a/b) = log(a) − log(b)
These properties allow multiplication to become addition — the foundation of the slide rule and logarithm tables.
Antilog
antilog₁₀(x) = 10^x
antilogₑ(x) = eˣ
The antilog is the inverse of the logarithm. antilog(3) = 10³ = 1000.

Frequently asked questions

What is the difference between log and ln?

log (common logarithm) uses base 10. ln (natural logarithm) uses base e ≈ 2.71828. In science and mathematics, ln is more common. In engineering and common tables, log base 10 is standard. log₁₀(x) = ln(x) / ln(10).

What is an antilogarithm?

An antilogarithm is the inverse operation of a logarithm. If log₁₀(x) = 2, then antilog₁₀(2) = 10² = 100. Essentially, antilog reverses the log operation by exponentiating the base.

Why can't I take the log of a negative number or zero?

Logarithms are only defined for positive real numbers. log(0) is undefined (approaches −∞). log of a negative number gives a complex result, which is outside basic real-number mathematics.

What is the change of base formula?

The change of base formula lets you compute any logarithm using the calculator's built-in log or ln: logᵦ(x) = log(x)/log(b) = ln(x)/ln(b). This is how calculators compute log base 2 or any custom base.

Where are logarithms used?

Logarithms are used in the Richter scale (earthquakes), decibels (sound intensity), pH scale (acidity), compound interest formulas, information theory (bits), and many areas of science and engineering where values span many orders of magnitude.