# Properties of Natural Logarithm

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## 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

- $\ln e = 1$

### Logarithm is Continuous

The real natural logarithm function is continuous.

### 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 $x \to 0^+$