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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:
: <math>a^x = e^{x \ln a}</math>
for all <i>a</i> > 0 and
The exponential function also gives rise to the [[trigonometric function]]s (as can be seen from [[Eulers formula in complex analysis|Euler's formula]]) and to the [[hyperbolic function]]s. Thus we see that all elementary functions except for the [[polynomial]]s spring from the exponential function in one way or another.
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