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

Calculus volume 1 Page 3 / 4

exp (ln x) = x for x > 0 and ln (exp x) = x for all x .

The following figure shows the graphs of $exp x$ and $ln x .$

This figure is a graph. It has three curves. The first curve is labeled exp x. It is an increasing curve with the x-axis as a horizontal asymptote. It intersects the y-axis at y=1. The second curve is a diagonal line through the origin. The third curve is labeled lnx. It is an increasing curve with the y-axis as an vertical axis. It intersects the x-axis at x=1. — The graphs of $ln x$ and $exp x .$

We hypothesize that $exp x = e^{x} .$ For rational values of $x,$ this is easy to show. If $x$ is rational, then we have $ln (e^{x}) = x ln e = x .$ Thus, when $x$ is rational, $e^{x} = exp x .$ For irrational values of $x,$ we simply define $e^{x}$ as the inverse function of $ln x .$

Definition

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

ln y = ln (e^{x}) = x .

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

e^{ln x} = x for x > 0 and ln (e^{x}) = 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

$e^{p} e^{q} = e^{p + q}$
$\frac{e^{p}}{e^{q}} = e^{p - q}$
${(e^{p})}^{r} = e^{p r}$

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

ln (e^{p} e^{q}) = ln (e^{p}) + ln (e^{q}) = p + q = ln (e^{p + q}) .

Since $ln x$ 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{matrix} ln y & = & x \\ \frac{d}{d x} ln y & = & \frac{d}{d x} x \\ \frac{1}{y} \frac{d y}{d x} & = & 1 \\ \frac{d y}{d x} & = & y . \end{matrix}

Thus, we see

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

as desired, which leads immediately to the integration formula

\int e^{x} d x = e^{x} + C .

We apply these formulas in the following examples.

Using properties of exponential functions

Evaluate the following derivatives:

$\frac{d}{d t} e^{3 t} e^{t^{2}}$
$\frac{d}{d x} e^{3 x^{2}}$

We apply the chain rule as necessary.

$\frac{d}{d t} e^{3 t} e^{t^{2}} = \frac{d}{d t} e^{3 t + t^{2}} = e^{3 t + t^{2}} (3 + 2 t)$
$\frac{d}{d x} e^{3 x^{2}} = e^{3 x^{2}} 6 x$

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Evaluate the following derivatives:

$\frac{d}{d x} (\frac{e^{x^{2}}}{e^{5 x}})$
$\frac{d}{d t} {(e^{2 t})}^{3}$

$\frac{d}{d x} (\frac{e^{x^{2}}}{e^{5 x}}) = e^{x^{2} - 5 x} (2 x - 5)$
$\frac{d}{d t} {(e^{2 t})}^{3} = 6 e^{6 t}$

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Using properties of exponential functions

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

Using $u$ -substitution, let $u = − x^{2} .$ Then $d u = −2 x d x,$ and we have

\int 2 x e^{− x^{2}} d x = − \int e^{u} d u = − e^{u} + C = − e^{− x^{2}} + C .

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

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

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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 $f (x) = a^{x} .$ 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 $f (x) = a^{x}$ 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 ln 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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Source: OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2

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