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Here <math>n!</math> stands for the [[factorial]] of <i>n</i> and <i>x</i> can be any [[real number|real]] or [[complex number|complex]] number, or even any element of a [[Banach algebra]] or the field of [[p-adic numbers|<i>p</i>-adic numbers]].
To see the equivalence of these definitions, see [[Definitions of the exponential function]].
If <i>x</i> is real, then <i>e<sup>x</sup></i> is positive and strictly increasing. Therefore its [[inverse function]], the [[natural logarithm]] ln(<i>x</i>), is defined for all positive <i>x</i>. Using the natural logarithm, one can define more general exponential functions as follows:
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