2.7 Integrals, exponential functions, and logarithms  (Page 3/4)

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$\text{exp}\left(\text{ln}\phantom{\rule{0.2em}{0ex}}x\right)=x\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}x>0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{ln}\left(\text{exp}\phantom{\rule{0.2em}{0ex}}x\right)=x\phantom{\rule{0.2em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}x.$

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 $\text{ln}\phantom{\rule{0.2em}{0ex}}x.$

Definition

For any real number $x,$ define $y={e}^{x}$ to be the number for which

$\text{ln}\phantom{\rule{0.2em}{0ex}}y=\text{ln}\left({e}^{x}\right)=x.$

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

${e}^{\text{ln}\phantom{\rule{0.2em}{0ex}}x}=x\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}x>0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{ln}\left({e}^{x}\right)=x$

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 ${e}^{x}.$

Properties of the exponential function

If $p$ and $q$ are any real numbers and $r$ is a rational number, then

1. ${e}^{p}{e}^{q}={e}^{p+q}$
2. $\frac{{e}^{p}}{{e}^{q}}={e}^{p-q}$
3. ${\left({e}^{p}\right)}^{r}={e}^{pr}$

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

$\text{ln}\left({e}^{p}{e}^{q}\right)=\text{ln}\left({e}^{p}\right)+\text{ln}\left({e}^{q}\right)=p+q=\text{ln}\left({e}^{p+q}\right).$

Since $\text{ln}\phantom{\rule{0.2em}{0ex}}x$ is one-to-one, then

${e}^{p}{e}^{q}={e}^{p+q}.$

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

$\begin{array}{ccc}\hfill \text{ln}\phantom{\rule{0.2em}{0ex}}y& =\hfill & x\hfill \\ \hfill \frac{d}{dx}\text{ln}\phantom{\rule{0.2em}{0ex}}y& =\hfill & \frac{d}{dx}x\hfill \\ \hfill \frac{1}{y}\phantom{\rule{0.1em}{0ex}}\frac{dy}{dx}& =\hfill & 1\hfill \\ \hfill \frac{dy}{dx}& =\hfill & y.\hfill \end{array}$

Thus, we see

$\frac{d}{dx}{e}^{x}={e}^{x}$

as desired, which leads immediately to the integration formula

$\int {e}^{x}dx={e}^{x}+C.$

We apply these formulas in the following examples.

Using properties of exponential functions

Evaluate the following derivatives:

1. $\frac{d}{dt}{e}^{3t}{e}^{{t}^{2}}$
2. $\frac{d}{dx}{e}^{3{x}^{2}}$

We apply the chain rule as necessary.

1. $\frac{d}{dt}{e}^{3t}{e}^{{t}^{2}}=\frac{d}{dt}{e}^{3t+{t}^{2}}={e}^{3t+{t}^{2}}\left(3+2t\right)$
2. $\frac{d}{dx}{e}^{3{x}^{2}}={e}^{3{x}^{2}}6x$

Evaluate the following derivatives:

1. $\frac{d}{dx}\left(\frac{{e}^{{x}^{2}}}{{e}^{5x}}\right)$
2. $\frac{d}{dt}{\left({e}^{2t}\right)}^{3}$
1. $\frac{d}{dx}\left(\frac{{e}^{{x}^{2}}}{{e}^{5x}}\right)={e}^{{x}^{2}-5x}\left(2x-5\right)$
2. $\frac{d}{dt}{\left({e}^{2t}\right)}^{3}=6{e}^{6t}$

Using properties of exponential functions

Evaluate the following integral: $\int 2x{e}^{\text{−}{x}^{2}}dx.$

Using $u$ -substitution, let $u=\text{−}{x}^{2}.$ Then $du=-2x\phantom{\rule{0.2em}{0ex}}dx,$ and we have

$\int 2x{e}^{\text{−}{x}^{2}}dx=\text{−}\int {e}^{u}du=\text{−}{e}^{u}+C=\text{−}{e}^{\text{−}{x}^{2}}+C.$

Evaluate the following integral: $\int \frac{4}{{e}^{3x}}dx.$

$\int \frac{4}{{e}^{3x}}dx=-\frac{4}{3}{e}^{-3x}+C$

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.

Definition

For any $a>0,$ and for any real number $x,$ define $y={a}^{x}$ as follows:

$y={a}^{x}={e}^{x\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}a}.$

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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