The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 100 to base 10 is 2, because 100 is 10 to the power 2: 1000 = 10 × 10 = 103. More generally, if x = by, then y is the logarithm of x to base b, and is written y = logb(x), so log10(100) = 2.
The inverse of a logarithmic function is an exponential functions.
Logarithms play a significant role in quantitative finance. If we use a base of natural e, we can compute continuously compounded returns.
The first video below provides an intuitive explanation of the concept of logarithms.
The video below discusses why we use log returns in finance.