The logarithm of a product of two numbers equals the sum of their logarithms:

This equation forms the basis for multiplying two numbers on a slide rule or using a logarithm table. Since adding is generally easier than multiplying, this led to the rapid adoption of logarithms for calculations after their invention by John Napier in the early 17th century. For such computational purposes, the logarithm to base b = 10 (common logarithm) was primarily used. The natural logarithm uses the constant e (approximately 2.718) as its base, and is especially widespread in calculus. The binary logarithm uses base b = 2 and primarily aids computing applications.

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Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the Richter scale uses the common logarithm to measure the amplitude of seismic events. Logarithms are commonplace in scientific formulas, measure the complexity of algorithms and of fractals, and appear in formulas counting prime numbers. They describe musical intervals, inform some models in psychophysics and can aid in forensic accounting.
The complex logarithm is the inverse of the exponential function applied to complex numbers and generalizes the logarithm to complex numbers. The discrete logarithm is another variant; it has applications in public-key cryptography.


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