What is Natural logarithm?

The exponential is continuous and strictly increasing from $\R$ onto $\intoo{0}{+\infty}$; by the bijection theorem (Theorem 21.15), for every $y > 0$ the equation $\eu^x = y$ has a unique solution. This solution is the natural logarithm of $y$, written $\ln y$. Thus

In particular $\ln 1 = 0$, $\ln \eu = 1$, and $\eu^{\ln y} = y$, $\ln(\eu^x) = x$.