The Exponential Function
We now turn our attention to the function \(e^{x} .\) Note that the natural logarithm is one-to-one and therefore has an inverse function. For now, we denote this inverse function by \(\text{exp} x .\) Then,
The following figure shows the graphs of \(\text{exp} x\) and \(\text{ln} x .\)
We hypothesize that \(\text{exp} x = e^{x} .\) For rational values of \(x ,\) this is easy to show. If \(x\) is rational, then we have \(\text{ln} \left(\right. e^{x} \left.\right) = x \text{ln} e = x .\) Thus, when \(x\) is rational, \(e^{x} = \text{exp} x .\) For irrational values of \(x ,\) we simply define \(e^{x}\) as the inverse function of \(\text{ln} x .\)
Definition
For any real number \(x ,\) define \(y = e^{x}\) to be the number for which
Then we have \(e^{x} = \text{exp} \left(\right. x \left.\right)\) for all \(x ,\) and thus
for all \(x .\)
This lesson is part of:
Applications of Integration