Exponential Functions

An exponential function is a function of the form

y=a^{x},

where a is a constant.

Figure 1 shows the graph of y=2^{x}, in blue, together with that of its gradient, in red. Notice that the function itself and the gradient are very similar to one another: they both rise rapidly as x gets larger.

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Figure 1: graph of y=2^{x} (blue line) and its gradient (red line)

Figure 2 shows y=3^{x} together with its gradient. Again, the two curves are very similar; this time, however, the gradient curve is just slightly above that of the function itself.

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Figure 2: graph of y=3^{x} (blue line) and its gradient (red line)

There is a number called e, lying between 2 and 3, such that the graph of y=e^{x} and that of its gradient are exactly the same.

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Figure 3: graph of y=e^{x}

Note that

e^{x+y}=e^{x}e^{y},
e^{x-y}=e^{x}/e^{y},
\left(e^{x}\right)^{n}=e^{n\,x}.
Created with the ExportAsWebPage package in Wolfram Mathematica 7.0