Logarithms explained simply

In short: A logarithm answers the question: “To what power must I raise a base to get a given value?” It is the inverse of exponentiation.

In more detail: log₂(8) = 3 because 2³ = 8. log₁₀(1000) = 3 because 10³ = 1000. The natural logarithm ln uses Euler’s number e ≈ 2.718 as its base.

In Depth

• Rules: log(a · b) = log a + log b (multiplication becomes addition), log(a / b) = log a − log b, log(aⁿ) = n · log a.
• Change of base: log_b(x) = ln(x) / ln(b). With a calculator you can compute any base this way.
• Why useful: The logarithm compresses huge ranges of numbers. On logarithmic scales powers of ten are equally far apart.
• Everyday examples: Decibel (loudness), Richter scale (earthquakes), pH value (acidity), star brightness, time axes in charts.
• Computing: The number of bits for n values is log₂(n). Binary search needs log₂(n) steps (binary search), sorting methods O(n log n) (Big-O notation).
• Inverse of exponential growth: How long until something doubles? ln(2)/ln(q) (exponential growth).
• Domain: Defined only for positive numbers. log(0) does not exist.

See also: SNR and decibel