# Properties of Natural Logarithm

## Theorem

Let $x \in \R$ be a real number such that $x > 0$.

Let $\ln x$ be the natural logarithm of $x$.

Then:

### Natural Logarithm of 1 is 0

$\ln 1 = 0$

### Natural Logarithm of e is 1

$\ln e = 1$

### Derivative of Natural Logarithm Function

Let $\ln x$ be the natural logarithm function.

Then:

$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$

### Logarithm is Strictly Increasing

$\ln x: x > 0$ is strictly increasing.

### Logarithm is Strictly Concave

$\ln x: x > 0$ strictly concave.

### Logarithm Tends to Infinity

$\ln x \to +\infty$ as $x \to +\infty$

### Logarithm Tends to Negative Infinity

$\ln x \to -\infty$ as $x \to 0^+$