Exponential function: Difference between revisions

Revision as of 13:07, 1 April 2003

Revision as of 13:14, 1 April 2003

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Here <math>n!</math> stands for the [[factorial]] of <math>n</math> and <math>x</math> 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]].

If ''x'' is real, then exp(''x'') is positive. It isand strictly increasing if x > 0. Therefore its [[inverse function]], the [[natural logarithm]] ln(''x''), is defined for all positive ''x''. Using the natural logarithm, one can define more general exponential functions as follows:

: <math>a^x = \exp(\ln(a) x)</math>

for all <math>a > 0</math> and all real <math>x</math>.