6.7 Integrals, exponential functions, and logarithms  (Page 2/4)

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 $\text{ln}\left(e^x\right)=x\text{ln}e=x.$ 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 $\text{ln}x.$

Then we have $e^x=\text{exp}\left(x\right)$ for all $x,$ and thus

for all $x.$

Properties of the exponential function

Since the exponential function was defined in terms of an inverse function, and not in terms of a power of $e,$ we must verify that the usual laws of exponents hold for the function $e^x.$

Proof

Note that if $p$ and $q$ are rational, the properties hold. However, if $p$ or $q$ are irrational, we must apply the inverse function definition of $e^x$ and verify the properties. Only the first property is verified here; the other two are left to you. We have

Since $\text{ln}x$ is one-to-one, then

□

As with part iv. of the logarithm properties, we can extend property iii. to irrational values of $r,$ and we do so by the end of the section.

We also want to verify the differentiation formula for the function $y=e^x.$ To do this, we need to use implicit differentiation. Let $y=e^x.$ Then

Thus, we see

as desired, which leads immediately to the integration formula

We apply these formulas in the following examples.

General logarithmic and exponential functions

We close this section by looking at exponential functions and logarithms with bases other than $e.$ Exponential functions are functions of the form $f\left(x\right)=a^x.$ Note that unless $a=e,$ we still do not have a mathematically rigorous definition of these functions for irrational exponents. Let’s rectify that here by defining the function $f\left(x\right)=a^x$ in terms of the exponential function $e^x.$ We then examine logarithms with bases other than $e$ as inverse functions of exponential functions.

Now $a^x$ is defined rigorously for all values of x . This definition also allows us to generalize property iv. of logarithms and property iii. of exponential functions to apply to both rational and irrational values of $r.$ It is straightforward to show that properties of exponents hold for general exponential functions defined in this way.