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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}}.
The exponential function is occasionally called the '''natural exponential function''', matching the name ''natural logarithm'', for distinguishing it from some other functions that are also commonly called ''exponential functions''. These functions include the functions of the form {{tmath|1=f(x) = b^x}}, which is [[exponentiation]] with a fixed base {{tmath|b}}. More generally, and especially in applications, functions of the general form {{tmath|1=f(x) = ab^x}} are also called exponential functions. They [[exponential growth|grow]] or [[exponential decay|decay]] exponentially in that the instantaneous rate that {{tmath|f(x)}} changes when {{tmath|x}} is increased is ''proportional'' to the current value of {{tmath|f(x)}}.
The exponential function can be generalized to accept [[complex number]]s as arguments. This reveals relations between multiplication of complex numbers, rotations in the [[complex plane]], and [[trigonometry]]. [[Euler's formula]] {{tmath|1= \exp i\theta = \cos\theta + i\sin\theta}} expresses and summarizes these relations.
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