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Common logarithm (log)

Calculate the base-10 logarithm. With graph and explanation.

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Co je common logarithm

The common logarithm log x is the logarithm with base 10. It tells you how many orders of magnitude (zeros) a number has. For example log 1000 = 3, because 1000 = 10³. Historically it was used to simplify multiplying large numbers.

Where it is used in practice

  • The pH scale in chemistry – acidity is the negative common logarithm of concentration.
  • Decibels (loudness) and the Richter scale (earthquake magnitude).
  • Number of digits: a number has ⌊log x⌋ + 1 digits.

Good to know

log 10 = 1, log 100 = 2, log 1000 = 3. Each increase of 1 on a logarithmic scale means ten times more – that is why a magnitude-7 earthquake is 10× stronger than a magnitude 6.

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.