Logarithms

What is a logarithm?

A logarithm is the inverse of exponentiation: it asks what power a base must be raised to in order to produce a given number.

In general, $\log_b x$ is the exponent you'd raise $b$ to, to get $x$. Try it yourself: $\log_6 1296 = \,?$ (since $6^4 = 1296$, the answer is 4).

Common, natural, and binary logarithms

The common logarithmuses base 10 (and is what's meant when no base is written): $\log_{10} 1000 = 3$, $\log_{10} 100 = 2$.

The natural logarithm, written $\ln x$, uses base $e \approx 2.71828$, and shows up naturally in continuous growth: the value of an investment under continuous compounding follows $FV = PV \cdot e^{it}$.

The binary logarithm uses base 2 and appears throughout computer science: $\log_2 16 = 4$, $\log_2 8 = 3$.

A logarithm of a number between 0 and 1 is negative (e.g. $\log_{10} 0.01 = -2$, since $10^{-2} = 0.01$), and the logarithm of zero or a negative number is undefined for real numbers, since no real power of a positive base ever produces zero or a negative result.

Logarithm rules

Since logarithms and exponentials with the same base are inverses of each other, they cancel: