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Antilogarithm

Antilogarithm – from a logarithm back to the number (aˣ).

Graf funkce

Co je antilogarithm

The antilogarithm is the inverse of the logarithm – from a logarithm value it returns the original number. If logₐx = y, then the antilogarithm is x = aʸ. It is simply raising the base to a power.

Where it is used in practice

  • Converting back from a logarithmic scale to ordinary numbers (from decibels to intensity, from pH to concentration).
  • Solving logarithmic equations – by exponentiating both sides.
  • Historically when calculating with logarithm tables and slide rules.

Good to know

The base-10 antilogarithm of y is 10ʸ, the base-e one is eʸ. It is just another name for a power of the base.

Logarithm and power are inverse operations

The logarithm is the inverse function of the power. Picturing it this way makes logarithms much easier to understand: logₐx = y ⟺ aʸ = x. For example log₂8 = 3, because 2³ = 8. The logarithm thus “unpacks” the exponent.

Rules of logarithms

logₐ(x · z) = logₐx + logₐz   (product → sum)
logₐ(x / z) = logₐx − logₐz   (quotient → difference)
logₐ(xʳ) = r · logₐx   (power → multiplication)
logₐx = log x / log a   (change of base)

In detail with examples: rules of logarithms.

Tip: a logarithm turns multiplication into addition – which is why logarithm tables and slide rules were once used to multiply large numbers quickly.