# Natural Logarithm of 1 is 0

(Redirected from 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:

$\ds \ln x = \int_1^x \frac {\d t} t$
$\ds \ln 1 = \int_1^1 \frac {\d t} t = 0$

$\blacksquare$

## Proof 2

We use the definition of the natural logarithm as the inverse of the exponential:

$\ln x = y \iff e^y = x$

Then:

 $\ds e^0$ $=$ $\ds 1$ Exponential of Zero $\ds \leadstoandfrom \ \$ $\ds \ln 1$ $=$ $\ds 0$

$\blacksquare$

## Proof 3

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:

 $\ds \ln 1$ $=$ $\ds \lim_{n \mathop \to \infty} n \paren {\sqrt [n] 1 - 1}$ $\ds$ $=$ $\ds \lim_{n \mathop \to \infty} n \times 0$ $\ds$ $=$ $\ds \lim_{n \mathop \to \infty} 0$ $\ds$ $=$ $\ds 0$

$\blacksquare$