Note that if we use the absolute value function and create a new function
$\text{ln}\phantom{\rule{0.2em}{0ex}}\left|x\right|,$ we can extend the domain of the natural logarithm to include
$x<0.$ Then
$\left(d\text{/}\left(dx\right)\right)\text{ln}\phantom{\rule{0.2em}{0ex}}\left|x\right|=1\text{/}x.$ This gives rise to the familiar integration formula.
Integral of (1/
u )
du
The natural logarithm is the antiderivative of the function
$f\left(u\right)=1\text{/}u\text{:}$
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.
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
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:
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,
Questions & Answers
where we get a research paper on Nano chemistry....?
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
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?
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?