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 :

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 to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x