Natural Logarithm of 1 is 0

Theorem

 * $\ln 1 = 0$

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

Proof 1
From the definition of natural logarithm:
 * $\displaystyle \ln x := \int_1^x \frac {\mathrm dt} t$

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

Proof 2
This depends on the definition of the natural logarithm as the inverse of the exponential function.

$e^x := \displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n$

The result follows from the exponential function being one-to-one.