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The following figure shows the graphs of $\text{exp}\phantom{\rule{0.2em}{0ex}}x$ and $\text{ln}\phantom{\rule{0.2em}{0ex}}x.$
We hypothesize that $\text{exp}\phantom{\rule{0.2em}{0ex}}x={e}^{x}.$ For rational values of $x,$ this is easy to show. If $x$ is rational, then we have $\text{ln}\left({e}^{x}\right)=x\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}e=x.$ Thus, when $x$ is rational, ${e}^{x}=\text{exp}\phantom{\rule{0.2em}{0ex}}x.$ For irrational values of $x,$ we simply define ${e}^{x}$ as the inverse function of $\text{ln}\phantom{\rule{0.2em}{0ex}}x.$
For any real number $x,$ define $y={e}^{x}$ to be the number for which
Then we have ${e}^{x}=\text{exp}\left(x\right)$ for all $x,$ and thus
for all $x.$
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 ${e}^{x}.$
If $p$ and $q$ are any real numbers and $r$ is a rational number, then
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 $\text{ln}\phantom{\rule{0.2em}{0ex}}x$ 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.
Evaluate the following derivatives:
We apply the chain rule as necessary.
Evaluate the following derivatives:
Evaluate the following integral: $\int 2x{e}^{\text{\u2212}{x}^{2}}dx}.$
Using $u$ -substitution, let $u=\text{\u2212}{x}^{2}.$ Then $du=\mathrm{-2}x\phantom{\rule{0.2em}{0ex}}dx,$ and we have
Evaluate the following integral: $\int \frac{4}{{e}^{3x}}}dx.$
$\int \frac{4}{{e}^{3x}}dx}=-\frac{4}{3}{e}^{\mathrm{-3}x}+C$
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.
For any $a>0,$ and for any real number $x,$ define $y={a}^{x}$ as follows:
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.
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