Exponential function: Difference between revisions

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In [[mathematics]], the '''exponential function''' is the unique [[real function]] which maps [[0|zero]] to [[1|one]] and has a [[derivative (mathematics)|derivative]] equal to its value (while being [[differentiable]] on all [[Real number|<math>\mathbb{R}</math>]]). The exponential of a variable {{tmath|x}} is denoted {{tmath|\exp x}} or {{tmath|e^x}}, with the two notations used interchangeably. It is called ''exponential'' because its argument can be seen as an [[exponent (mathematics)|exponent]] to which a constant [[e (mathematical constant)|number {{math|''e'' ≈ 2.718}}]], the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature.

The exponential function converts sums to products: it maps the [[additive identity]] {{math|0}} to the [[multiplicative identity]] {{math|1}}, and the exponential of a sum is equal to the product of separate exponentials, {{tmath|1=\exp(x + y) = \exp x \cdot \exp y }}. Its [[inverse function]], the [[natural logarithm]], {{tmath|\ln}} or {{tmath|\log}}, converts products to sums: {{tmath|1= \ln(x\cdot y) = \ln x + \ln y}}.