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Common logarithms

The common logarithm says which exponent 10 must have to give the number. The logarithm of 10 to the n is n, and the logarithm of a number between 0 and 1 is negative. You can only take the logarithm of positive numbers. The rules turn products into sums and quotients into differences, and a power becomes a factor in front of the logarithm. That is why logarithms solve exponential equations.

log⁡10(10n)=n\log_{10}(10^n) = nthe definition
log⁡10(ab)=log⁡10a+log⁡10b\log_{10}(ab) = \log_{10} a + \log_{10} bproduct rule

Symbols

nnthe exponent

Example

log⁡101000=3\log_{10} 1000 = 3, because 103=100010^3 = 1000.

log⁡100.01=−2\log_{10} 0.01 = -2, because 10−2=0.0110^{-2} = 0.01.

The argument must be positive. The logarithm of a negative number does not exist.
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Part of Mathematics S1: Logarithms and exponential equations.