# 3.10 Domain and range of exponential and logarithmic function

 Page 1 / 2

Working rules : We shall be using following definitions/results for solving problems in this module :

• $y=\mathrm{log}{}_{a}x,\phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}a>0,a\ne 1,x>0,y\in R$
• $y={\mathrm{log}}_{a}x⇔x={a}^{y}$
• $\text{If}\phantom{\rule{1em}{0ex}}{\mathrm{log}}_{a}x\ge y,\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}x\ge {a}^{y},\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}a>1$
• $\text{If}\phantom{\rule{1em}{0ex}}\mathrm{log}{}_{a}x\ge y,\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}x\le {a}^{y},\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}a<1$

## Domain of different logarithmic functions

Problem : Find the domain of the function given by (Be aware that "x" appears as base of given logrithmic function):

$f\left(x\right)={\mathrm{log}}_{x}2$

Solution : By definition of logarithmic function, we know that base of logarithmic function is a positive number excluding x =1.

$x>0,\phantom{\rule{1em}{0ex}}x\ne 1$

Hence, domain of the given function is :

$\text{Domain}=\left(0,\infty \right)-\left\{1\right\}$

or,

$\text{Domain}=\left(0,1\right)\cup \left\{1,\infty \right\}$

Problem : Find the domain of the function given by :

$f\left(x\right)={\mathrm{log}}_{10}\frac{{x}^{2}-5x+6}{{x}^{2}+5x+9}$

Solution : The argument (input to the function) of logarithmic function is a rational function. We need to find values of “x” such that the argument of the function evaluates to a positive number. Hence,

$⇒\frac{{x}^{2}-5x+6}{{x}^{2}+5x+9}>0$

In this case, we can not apply sign scheme for the rational function as a whole. Reason is that the quadratic equation in the denominator has no real roots and as such can not be factorized in linear factors. We see that discreminant,"D", of the quadratic equation in the denominator, is negative :

$⇒D={b}^{2}-4ac={5}^{2}-4X1X9=25-36=-11$

The quadratic expression in denominator is positive for all value of x as coefficient of squared term is positive. Clearly, sign of rational function is same as that of quadratic expression in the numerator. The coefficient of squared term of the numerator “ ${x}^{2}$ ”, is positive for all values of “x”. The quadratic expression in the numerator evaluates to positive for intervals beyond root values. The roots of the corresponding equal equation is :

$⇒{x}^{2}-2x-3x+6=0\phantom{\rule{1em}{0ex}}⇒x\left(x-2\right)-3\left(x-2\right)=0\phantom{\rule{1em}{0ex}}⇒\left(x-2\right)\left(x-3\right)=0$

$x<2\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x>3$

$\text{Domain}=\left(-\infty ,2\right)\phantom{\rule{1em}{0ex}}\cup \phantom{\rule{1em}{0ex}}\left(3,\infty \right)$

Problem : Find the domain of the function given by :

$f\left(x\right)=\sqrt{\mathrm{log}{}_{10}\frac{6x-{x}^{2}}{8}}$

Solution : The function is a square root of a logarithmic function. On the other hand argument of logarithmic function is a rational function. In order to find the domain of the given function, we first determine what values of “x” are valid for logarithmic function. Then, we apply the condition that expression within square root should be non-negative number. Domain of given function is intersection of intervals of x obtained for each of these conditions. Now, we know that argument (input to function) of logarithmic function is a positive number. This implies that we need to find the interval of “x” for which,

$⇒\frac{6x-{x}^{2}}{8}>0$

$⇒6x-{x}^{2}>0$

In above step, we should emphasize here that we multiply “8” and “0” and retain the inequality sign because 8>0. Now, we multiply the inequality by “-1”. Therefore, inequality sign is reversed.

$⇒{x}^{2}-6x<0$

Here, roots of corresponding quadratic equation “ ${x}^{2}-6x$ ” is x = 0, 6. It means that middle interval between “0 and 6” is negative as coefficient of “ ${x}^{2}$ ” is positive i.e. 6>0. Hence, interval satisfying the inequality is :

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
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
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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x