Natural Logarithm of 1 is 0/Proof 3

Theorem

$\ln 1 = 0$

Proof

We use the definition of the natural logarithm as the limit of a sequence:

$\ds \ln x = \lim_{n \mathop \to \infty} n \paren {\sqrt [n] x - 1}$

Then:

\(\ds \ln 1\)

\(=\)

\(\ds \lim_{n \mathop \to \infty} n \paren {\sqrt [n] 1 - 1}\)

\(\ds \)

\(=\)

\(\ds \lim_{n \mathop \to \infty} n \times 0\)

\(\ds \)

\(=\)

\(\ds \lim_{n \mathop \to \infty} 0\)

\(\ds \)

\(=\)

\(\ds 0\)

$\blacksquare$