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## Definitions

### General

The natural logarithm of a number x, is a power p so that e^{p} = x, where e is a mathematical constant, approximately equal to 2.718. In modern notation natural logarithm is written as \(\ln{x}\) or \(\log_{e}{x}\). Any positive real number has a natural logarithm. The natural logarithm of 1 is 0, while the natural logarithm of 0 approaches negative infinity.

The graph of the natural logarithm function is shown in the figure below. It is a monotonic function.

### Properties

The derivative of the natural logarithm function is:

\[ \left(\ln x\right)' = \frac{1}{x} \]

The integral of the natural logarithm function is given by:

\[ \int \ln x\, \mathrm{d}x = x\ln{x}-x +C \]