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

 Page 2 / 4

## Calculating derivatives of natural logarithms

Calculate the following derivatives:

1. $\frac{d}{dx}\text{ln}\left(5{x}^{3}-2\right)$
2. $\frac{d}{dx}{\left(\text{ln}\left(3x\right)\right)}^{2}$

We need to apply the chain rule in both cases.

1. $\frac{d}{dx}\text{ln}\left(5{x}^{3}-2\right)=\frac{15{x}^{2}}{5{x}^{3}-2}$
2. $\frac{d}{dx}{\left(\text{ln}\left(3x\right)\right)}^{2}=\frac{2\left(\text{ln}\left(3x\right)\right)·3}{3x}=\frac{2\left(\text{ln}\left(3x\right)\right)}{x}$

Calculate the following derivatives:

1. $\frac{d}{dx}\text{ln}\left(2{x}^{2}+x\right)$
2. $\frac{d}{dx}{\left(\text{ln}\left({x}^{3}\right)\right)}^{2}$
1. $\frac{d}{dx}\text{ln}\left(2{x}^{2}+x\right)=\frac{4x+1}{2{x}^{2}+x}$
2. $\frac{d}{dx}{\left(\text{ln}\left({x}^{3}\right)\right)}^{2}=\frac{6\phantom{\rule{0.2em}{0ex}}\text{ln}\left({x}^{3}\right)}{x}$

Note that if we use the absolute value function and create a new function $\text{ln}\phantom{\rule{0.2em}{0ex}}|x|,$ we can extend the domain of the natural logarithm to include $x<0.$ Then $\left(d\text{/}\left(dx\right)\right)\text{ln}\phantom{\rule{0.2em}{0ex}}|x|=1\text{/}x.$ This gives rise to the familiar integration formula.

## Integral of (1/ u ) du

The natural logarithm is the antiderivative of the function $f\left(u\right)=1\text{/}u\text{:}$

$\int \frac{1}{u}du=\text{ln}\phantom{\rule{0.2em}{0ex}}|u|+C.$

## Calculating integrals involving natural logarithms

Calculate the integral $\int \frac{x}{{x}^{2}+4}dx.$

Using $u$ -substitution, let $u={x}^{2}+4.$ Then $du=2x\phantom{\rule{0.2em}{0ex}}dx$ and we have

$\int \frac{x}{{x}^{2}+4}dx=\frac{1}{2}\int \frac{1}{u}du\frac{1}{2}\text{ln}\phantom{\rule{0.2em}{0ex}}|u|+C=\frac{1}{2}\text{ln}\phantom{\rule{0.2em}{0ex}}|{x}^{2}+4|+C=\frac{1}{2}\text{ln}\left({x}^{2}+4\right)+C.$

Calculate the integral $\int \frac{{x}^{2}}{{x}^{3}+6}dx.$

$\int \frac{{x}^{2}}{{x}^{3}+6}dx=\frac{1}{3}\text{ln}\phantom{\rule{0.2em}{0ex}}|{x}^{3}+6|+C$

Although we have called our function a “logarithm,” we have not actually proved that any of the properties of logarithms hold for this function. We do so here.

## Properties of the natural logarithm

If $a,b>0$ and $r$ is a rational number, then

1. $\text{ln}\phantom{\rule{0.2em}{0ex}}1=0$
2. $\text{ln}\left(ab\right)=\text{ln}\phantom{\rule{0.2em}{0ex}}a+\text{ln}\phantom{\rule{0.2em}{0ex}}b$
3. $\text{ln}\left(\frac{a}{b}\right)=\text{ln}\phantom{\rule{0.2em}{0ex}}a-\text{ln}\phantom{\rule{0.2em}{0ex}}b$
4. $\text{ln}\left({a}^{r}\right)=r\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}a$

## Proof

1. By definition, $\text{ln}\phantom{\rule{0.2em}{0ex}}1={\int }_{1}^{1}\frac{1}{t}dt=0.$
2. We have
$\text{ln}\left(ab\right)={\int }_{1}^{ab}\frac{1}{t}dt={\int }_{1}^{a}\frac{1}{t}dt+{\int }_{a}^{ab}\frac{1}{t}dt.$

Use $u\text{-substitution}$ on the last integral in this expression. Let $u=t\text{/}a.$ Then $du=\left(1\text{/}a\right)dt.$ Furthermore, when $t=a,u=1,$ and when $t=ab,u=b.$ So we get
$\text{ln}\left(ab\right)={\int }_{1}^{a}\frac{1}{t}dt+{\int }_{a}^{ab}\frac{1}{t}dt={\int }_{1}^{a}\frac{1}{t}dt+{\int }_{1}^{ab}\frac{a}{t}·\frac{1}{a}dt={\int }_{1}^{a}\frac{1}{t}dt+{\int }_{1}^{b}\frac{1}{u}du=\text{ln}\phantom{\rule{0.2em}{0ex}}a+\text{ln}\phantom{\rule{0.2em}{0ex}}b.$
3. Note that
$\frac{d}{dx}\text{ln}\left({x}^{r}\right)=\frac{r{x}^{r-1}}{{x}^{r}}=\frac{r}{x}.$

Furthermore,
$\frac{d}{dx}\left(r\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}x\right)=\frac{r}{x}.$

Since the derivatives of these two functions are the same, by the Fundamental Theorem of Calculus, they must differ by a constant. So we have
$\text{ln}\left({x}^{r}\right)=r\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}x+C$

for some constant $C.$ Taking $x=1,$ we get
$\begin{array}{ccc}\hfill \text{ln}\left({1}^{r}\right)& =\hfill & r\phantom{\rule{0.2em}{0ex}}\text{ln}\left(1\right)+C\hfill \\ \hfill 0& =\hfill & r\left(0\right)+C\hfill \\ \hfill C& =\hfill & 0.\hfill \end{array}$

Thus $\text{ln}\left({x}^{r}\right)=r\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}x$ and the proof is complete. Note that we can extend this property to irrational values of $r$ later in this section.
Part iii. follows from parts ii. and iv. and the proof is left to you.

## Using properties of logarithms

Use properties of logarithms to simplify the following expression into a single logarithm:

$\text{ln}\phantom{\rule{0.2em}{0ex}}9-2\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}3+\text{ln}\left(\frac{1}{3}\right).$

We have

$\text{ln}\phantom{\rule{0.2em}{0ex}}9-2\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}3+\text{ln}\left(\frac{1}{3}\right)=\text{ln}\left({3}^{2}\right)-2\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}3+\text{ln}\left({3}^{-1}\right)=2\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}3-2\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}3-\text{ln}\phantom{\rule{0.2em}{0ex}}3=\text{−}\text{ln}\phantom{\rule{0.2em}{0ex}}3.$

Use properties of logarithms to simplify the following expression into a single logarithm:

$\text{ln}\phantom{\rule{0.2em}{0ex}}8-\text{ln}\phantom{\rule{0.2em}{0ex}}2-\text{ln}\left(\frac{1}{4}\right).$

$4\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}2$

## Defining the number e

Now that we have the natural logarithm defined, we can use that function to define the number $e.$

## Definition

The number $e$ is defined to be the real number such that

$\text{ln}\phantom{\rule{0.2em}{0ex}}e=1.$

To put it another way, the area under the curve $y=1\text{/}t$ between $t=1$ and $t=e$ is $1$ ( [link] ). The proof that such a number exists and is unique is left to you. ( Hint : Use the Intermediate Value Theorem to prove existence and the fact that $\text{ln}\phantom{\rule{0.2em}{0ex}}x$ is increasing to prove uniqueness.)

The number $e$ can be shown to be irrational, although we won’t do so here (see the Student Project in Taylor and Maclaurin Series ). Its approximate value is given by

$e\approx 2.71828182846.$

## The exponential function

We now turn our attention to the function ${e}^{x}.$ Note that the natural logarithm is one-to-one and therefore has an inverse function. For now, we denote this inverse function by $\text{exp}\phantom{\rule{0.2em}{0ex}}x.$ Then,

where we get a research paper on Nano chemistry....?
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
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
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