Natural Logarithm of 1 is 0

Theorem

 * $\ln 1 = 0$

where $\ln 1$ denotes the natural logarithm of $1$.

Proof 1
We use the definition of the natural logarithm as an integral:
 * $\displaystyle \ln x = \int_1^x \frac {\mathrm dt} t$

From Integral on Zero Interval:
 * $\displaystyle \ln 1 = \int_1^1 \frac {\mathrm dt} t = 0$

Proof 2
We use the definition of the natural logarithm as the inverse of the exponential:
 * $\ln x = y \iff e^y = x$