Question

How does a value increase logarithmically?

Answer 1

LogarithmicallyIncreasingFunction
A function whose value increases more slowly to infinity than any nonconstant polynomial is said to be a logarithmically increasing function. The prototypical example is the function lnx, plotted above

Answer 2

Logarithmic growth is the inverse of exponential growth and is very slow. A familiar example of logarithmic growth is the number of digits needed to represent a number, N, in positional notation, which grows as logb (N), where b is the base of the number system used, e.g. 10 for decimal arithmetic.