Logarithm Calculator

What does the logarithm calculator do?

It has four tabs. The calculate tab finds a logarithm in any base: when the result is exact (log₂ 8 = 3, log₄ 8 = 3/2) it is given as a fraction, otherwise as a decimal. The unknown tab finds x, b or y in log_b x = y. The change of base tab rewrites a logarithm as a quotient of logarithms in another base. The rules tab expands one logarithm or condenses several into one.

What is a logarithm?

log_b x = y means that b raised to the power y gives x: b^y = x. Example: log₂ 8 = 3 because 2³ = 8. The base must be positive and different from 1, and the argument must be positive. The logarithm in base 10 is written log and the one in base e is written ln. In the base box you can type 10, e, 2 or any other positive number.

Change of base

Calculators only have log and ln. A logarithm in another base is found with the rule log_b x = ln x / ln b. Example: log₂ 10 = ln 10 / ln 2 = 2.302585 / 0.693147 = 3.321928.

Antilogarithm and unknown base

If x is the unknown in log_b x = y, then x = b^y. This is the antilogarithm: if log x = −2 then x = 10⁻² = 0.01. If the base is the unknown, b = x^(1/y): if log_b 81 = 4 then b = 81^(1/4) = 3.

Logarithm rules

• Product: log(a·b) = log a + log b
• Quotient: log(a/b) = log a − log b
• Power: log(aⁿ) = n·log a
• Root: log(√a) = (1/2)·log a

Expanding: log(x^2*y/z) = 2log(x) + log(y) − log(z). Condensing: 2log(x) + 3log(y) - log(z) = log(x²·y³/z). In the rules tab type log (base 10), ln (base e) or log2 (base 2). There is no rule for the logarithm of a sum: log(x + y) cannot be expanded.

Limits

The logarithm of zero or of a negative number is undefined in the real numbers. Decimal results are rounded to 12 significant digits. An exact result is looked for only when the number and the base are whole numbers or fractions and the exponent is a fraction with a small denominator. Expanding and condensing are valid when every expression inside a logarithm is positive. When condensing, all logarithms must have the same base. For solving general logarithmic equations see the equation solver.