# 3.9 Exponential and logarithmic functions  (Page 2/3)

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The nature of exponential function are different around a=1. The plots of exponential functions for two cases (i)0<a<1 and (ii) a>1 are discussed here. If the base is greater than zero, but less than “1”, then the exponential function asymptotes to positive x-axis. It is easy to visualize the nature of plot. It is placed in the positive upper part as "f(x)" is positive. Also, note that ${\left(0.25\right)}^{2}$ is greater than ( ${\left(0.25\right)}^{4}$ . Hence, plot begins from a higher value to lower value as "x" increases, but never becomes equal to zero.

If the base is greater than “1”, then the exponential function asymptotes to negative x-axis. Again, it is easy to visualize the nature of plot. It is placed in the positive upper part as "f(x)" is positive. Also, note that ${\left(1.25\right)}^{2}$ is less than ( ${\left(1.25\right)}^{4}$ . Hence, plot begins from a lower value to higher value as "x" increases.

Note that expanse of exponential function is along x – axis on either side of the y-axis, showing that its domain is R. On the other hand, the expanse of “y” is limited to positive side of y-axis, showing that its range is positive real number. Further, irrespective of base values, all plots intersect y-axis at the same point i.e. y = 1 as :

$y={a}^{x}={a}^{0}=1$

## Logarithmic functions

A logarithmic function gives “exponent” of an expression in terms of a base, “a”, and a number, “x”. The following two representations, in this context, are equivalent :

${a}^{y}=x$

and

$f\left(x\right)=y={\mathrm{log}}_{a}x$

where :

• The base “a” is positive real number, but excluding “1”. Symbolically, $a>0,a\ne 1$ .
• The number “x” represents result of exponentiation, “ ${a}^{y}$ ” and is also a positive real number. Symbolically, x>0.
• The exponent “y” i.e. logarithm of “x” is a real number.

Note that neither “a” nor “x” equals to zero.

The expression of a logarithm for “x” on a certain base represents logarithmic function. In words, we can say that a logarithmic function associates every positive real number (x) to a real valued exponent (y), which is symbolically represented as :

$f\left(x\right)=y=\mathrm{log}{}_{a}x;\phantom{\rule{1em}{0ex}}a,x>0,\phantom{\rule{1em}{0ex}}a\ne 1$

Following earlier discussion for the case of exponential function, we exclude "a = 1" as logarithmic function is not relevant to this base.

${1}^{y}=1$

We can easily see here that whatever be the exponent, the value of logarithmic function is “1". Hence, base “1” is irrelevant as exponent “y” is not uniquely associated with “x”.

From the defining values of "x" and "f(x)", we conclude that domain and range of logarithmic function is :

$\text{Value of “x”}=\text{Domain}=\left(0,\infty \right)$

$\text{Value of “y”}=\text{Range}=R$

Note that domain and range of logarithmic function is exchanged with respect to domain and range of exponential function.

## Base

The base of the logarithmic function can be any positive number. However, “10” and “e” are two common bases that we often use. Here, “e” is a mathematical constant given by :

$e=2.718281828$

If we use “e” as the base, then the corresponding logarithmic function is called “natural” logarithmic function. The plots, here, show logarithmic functions for two bases (i) 10 and (ii) e.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
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
has a lot of application modern world
Kamaluddeen
yes
narayan
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x

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