TAHA Engineering Toolsby TAHA Knowledge Lab

Mathematics & Statistics

Logarithm Calculator

log of x to base 10, e, 2 or any base; antilog b^y; or the base b for given x and y.

Inputs

Calculate from
Logarithm curve rising slowly through the points (1, 0) and (b, 1), and the exponential curve b to the power x, mirror images about the dashed line y equals x.xyy = xy = bˣy = log_b x(1, 0)(b, 1)
The logarithm y = log_b x is the inverse of the power x = b^y: the two curves are mirror images in the line y = x. Every log curve passes through (1, 0) and (b, 1); it exists only for x > 0. Diagram not to scale.

Result

log_b(x)
3
ln x (base e)
6.90775527898
log₁₀ x
3
log₂ x
9.96578428466
Working
log_10(1,000) = ln 1,000 ÷ ln 10 = 3 Check: 10^3 = 1,000

The logarithm to base 10 of 1,000 is 3.

Educational estimate — check the assumptions below.

About this calculator

A logarithm answers “to what power must the base be raised to give this number?”: log₁₀ 1000 = 3 because 10³ = 1000. Engineers meet logarithms in decibels, pH, the Richter scale, exponential decay and growth, and log-scale charts. This calculator finds the logarithm to base 10, e, 2 or any base, the antilog (the power by), or the base for a given number and exponent.

Formula

log_b(x)

y = logb x  ⇔  by = x    logb x = ln x ÷ ln b

Rules: log(xy) = log x + log y; log(x/y) = log x − log y; log(xn) = n log x. x > 0, b > 0, b ≠ 1.

Antilog b^y

x = by (antilog of y to base b)

Find the base b

b = x1/y   (from by = x)

Worked example

  • log₁₀ 1000 = 3; ln 1000 = 6.90775527898; log₂ 1000 = 9.96578428466
  • log₅ 125 = ln 125 ÷ ln 5 = 3
  • Antilog: 10³ = 1000; e² = 7.38905609893
  • Base: b³ = 1000 → b = 10001/3 = 10

Assumptions and limitations

  • Real logarithms only: x must be greater than 0, and the base positive and not 1.
  • Bases 10, e and 2 use the built-in functions directly; another base uses ln x ÷ ln b, which can differ from an exact integer in the last digit (shown rounded to 12 significant figures).

References

  • Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions, chapter 4 (elementary transcendental functions).