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

 Page 3 / 4
$\text{exp}\left(\text{ln}\phantom{\rule{0.2em}{0ex}}x\right)=x\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}x>0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{ln}\left(\text{exp}\phantom{\rule{0.2em}{0ex}}x\right)=x\phantom{\rule{0.2em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}x.$

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.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul

#### Get Jobilize Job Search Mobile App in your pocket Now! By OpenStax By Sandhills MLT By Rylee Minllic By Yacoub Jayoghli By OpenStax By Stephen Voron By Cath Yu By Robert Murphy By Brooke Delaney By