# 4.6 Exponential and logarithmic equations  (Page 7/8)

 Page 7 / 8

When does an extraneous solution occur? How can an extraneous solution be recognized?

When can the one-to-one property of logarithms be used to solve an equation? When can it not be used?

The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base.

## Algebraic

For the following exercises, use like bases to solve the exponential equation.

${4}^{-3v-2}={4}^{-v}$

$64\cdot {4}^{3x}=16$

$x=-\frac{1}{3}$

${3}^{2x+1}\cdot {3}^{x}=243$

${2}^{-3n}\cdot \frac{1}{4}={2}^{n+2}$

$n=-1$

$625\cdot {5}^{3x+3}=125$

$\frac{{36}^{3b}}{{36}^{2b}}={216}^{2-b}$

$b=\frac{6}{5}$

${\left(\frac{1}{64}\right)}^{3n}\cdot 8={2}^{6}$

For the following exercises, use logarithms to solve.

${9}^{x-10}=1$

$x=10$

$2{e}^{6x}=13$

${e}^{r+10}-10=-42$

No solution

$2\cdot {10}^{9a}=29$

$-8\cdot {10}^{p+7}-7=-24$

$p=\mathrm{log}\left(\frac{17}{8}\right)-7$

$7{e}^{3n-5}+5=-89$

${e}^{-3k}+6=44$

$k=-\frac{\mathrm{ln}\left(38\right)}{3}$

$-5{e}^{9x-8}-8=-62$

$-6{e}^{9x+8}+2=-74$

$x=\frac{\mathrm{ln}\left(\frac{38}{3}\right)-8}{9}$

${2}^{x+1}={5}^{2x-1}$

${e}^{2x}-{e}^{x}-132=0$

$7{e}^{8x+8}-5=-95$

$10{e}^{8x+3}+2=8$

$x=\frac{\mathrm{ln}\left(\frac{3}{5}\right)-3}{8}$

$4{e}^{3x+3}-7=53$

$8{e}^{-5x-2}-4=-90$

no solution

${3}^{2x+1}={7}^{x-2}$

${e}^{2x}-{e}^{x}-6=0$

$x=\mathrm{ln}\left(3\right)$

$3{e}^{3-3x}+6=-31$

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.

$\mathrm{log}\left(\frac{1}{100}\right)=-2$

${10}^{-2}=\frac{1}{100}$

${\mathrm{log}}_{324}\left(18\right)=\frac{1}{2}$

For the following exercises, use the definition of a logarithm to solve the equation.

$5{\mathrm{log}}_{7}n=10$

$n=49$

$-8{\mathrm{log}}_{9}x=16$

$4+{\mathrm{log}}_{2}\left(9k\right)=2$

$k=\frac{1}{36}$

$2\mathrm{log}\left(8n+4\right)+6=10$

$10-4\mathrm{ln}\left(9-8x\right)=6$

$x=\frac{9-e}{8}$

For the following exercises, use the one-to-one property of logarithms to solve.

$\mathrm{ln}\left(10-3x\right)=\mathrm{ln}\left(-4x\right)$

${\mathrm{log}}_{13}\left(5n-2\right)={\mathrm{log}}_{13}\left(8-5n\right)$

$n=1$

$\mathrm{log}\left(x+3\right)-\mathrm{log}\left(x\right)=\mathrm{log}\left(74\right)$

$\mathrm{ln}\left(-3x\right)=\mathrm{ln}\left({x}^{2}-6x\right)$

No solution

${\mathrm{log}}_{4}\left(6-m\right)={\mathrm{log}}_{4}3m$

$\mathrm{ln}\left(x-2\right)-\mathrm{ln}\left(x\right)=\mathrm{ln}\left(54\right)$

No solution

${\mathrm{log}}_{9}\left(2{n}^{2}-14n\right)={\mathrm{log}}_{9}\left(-45+{n}^{2}\right)$

$\mathrm{ln}\left({x}^{2}-10\right)+\mathrm{ln}\left(9\right)=\mathrm{ln}\left(10\right)$

$x=±\frac{10}{3}$

For the following exercises, solve each equation for $\text{\hspace{0.17em}}x.$

$\mathrm{log}\left(x+12\right)=\mathrm{log}\left(x\right)+\mathrm{log}\left(12\right)$

$\mathrm{ln}\left(x\right)+\mathrm{ln}\left(x-3\right)=\mathrm{ln}\left(7x\right)$

$x=10$

${\mathrm{log}}_{2}\left(7x+6\right)=3$

$\mathrm{ln}\left(7\right)+\mathrm{ln}\left(2-4{x}^{2}\right)=\mathrm{ln}\left(14\right)$

$x=0$

${\mathrm{log}}_{8}\left(x+6\right)-{\mathrm{log}}_{8}\left(x\right)={\mathrm{log}}_{8}\left(58\right)$

$\mathrm{ln}\left(3\right)-\mathrm{ln}\left(3-3x\right)=\mathrm{ln}\left(4\right)$

$x=\frac{3}{4}$

${\mathrm{log}}_{3}\left(3x\right)-{\mathrm{log}}_{3}\left(6\right)={\mathrm{log}}_{3}\left(77\right)$

## Graphical

For the following exercises, solve the equation for $\text{\hspace{0.17em}}x,$ if there is a solution . Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.

${\mathrm{log}}_{9}\left(x\right)-5=-4$

$x=9$

${\mathrm{log}}_{3}\left(x\right)+3=2$

$\mathrm{ln}\left(3x\right)=2$

$x=\frac{{e}^{2}}{3}\approx 2.5$

$\mathrm{ln}\left(x-5\right)=1$

$\mathrm{log}\left(4\right)+\mathrm{log}\left(-5x\right)=2$

$x=-5$

$-7+{\mathrm{log}}_{3}\left(4-x\right)=-6$

$\mathrm{ln}\left(4x-10\right)-6=-5$

$x=\frac{e+10}{4}\approx 3.2$

$\mathrm{log}\left(4-2x\right)=\mathrm{log}\left(-4x\right)$

${\mathrm{log}}_{11}\left(-2{x}^{2}-7x\right)={\mathrm{log}}_{11}\left(x-2\right)$

No solution

$\mathrm{ln}\left(2x+9\right)=\mathrm{ln}\left(-5x\right)$

${\mathrm{log}}_{9}\left(3-x\right)={\mathrm{log}}_{9}\left(4x-8\right)$

$x=\frac{11}{5}\approx 2.2$

$\mathrm{log}\left({x}^{2}+13\right)=\mathrm{log}\left(7x+3\right)$

$\frac{3}{{\mathrm{log}}_{2}\left(10\right)}-\mathrm{log}\left(x-9\right)=\mathrm{log}\left(44\right)$

$x=\frac{101}{11}\approx 9.2$

$\mathrm{ln}\left(x\right)-\mathrm{ln}\left(x+3\right)=\mathrm{ln}\left(6\right)$

For the following exercises, solve for the indicated value, and graph the situation showing the solution point.

An account with an initial deposit of $\text{\hspace{0.17em}}\text{6,500}\text{\hspace{0.17em}}$ earns $\text{\hspace{0.17em}}7.25%\text{\hspace{0.17em}}$ annual interest, compounded continuously. How much will the account be worth after 20 years?

about $\text{\hspace{0.17em}}27,710.24$

The formula for measuring sound intensity in decibels $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ is defined by the equation $\text{\hspace{0.17em}}D=10\mathrm{log}\left(\frac{I}{{I}_{0}}\right),\text{}$ where $\text{\hspace{0.17em}}I\text{\hspace{0.17em}}$ is the intensity of the sound in watts per square meter and $\text{\hspace{0.17em}}{I}_{0}={10}^{-12}\text{\hspace{0.17em}}$ is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of $\text{\hspace{0.17em}}8.3\cdot {10}^{2}\text{\hspace{0.17em}}$ watts per square meter?

The population of a small town is modeled by the equation $\text{\hspace{0.17em}}P=1650{e}^{0.5t}\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in years. In approximately how many years will the town’s population reach $\text{\hspace{0.17em}}\text{20,000?}$

## Technology

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to 3 decimal places .

$1000{\left(1.03\right)}^{t}=5000\text{\hspace{0.17em}}$ using the common log.

${e}^{5x}=17\text{\hspace{0.17em}}$ using the natural log

$\frac{\mathrm{ln}\left(17\right)}{5}\approx 0.567$

$3{\left(1.04\right)}^{3t}=8\text{\hspace{0.17em}}$ using the common log

${3}^{4x-5}=38\text{\hspace{0.17em}}$ using the common log

$50{e}^{-0.12t}=10\text{\hspace{0.17em}}$ using the natural log

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

$7{e}^{3x-5}+7.9=47$

$x\approx 2.2401$

$\mathrm{ln}\left(3\right)+\mathrm{ln}\left(4.4x+6.8\right)=2$

$\mathrm{log}\left(-0.7x-9\right)=1+5\mathrm{log}\left(5\right)$

$x\approx -\text{44655}.\text{7143}$

Atmospheric pressure $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ in pounds per square inch is represented by the formula $\text{\hspace{0.17em}}P=14.7{e}^{-0.21x},$ where $x$ is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of $\text{\hspace{0.17em}}8.369\text{\hspace{0.17em}}$ pounds per square inch? ( Hint : there are 5280 feet in a mile)

The magnitude M of an earthquake is represented by the equation $\text{\hspace{0.17em}}M=\frac{2}{3}\mathrm{log}\left(\frac{E}{{E}_{0}}\right)\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}E\text{\hspace{0.17em}}$ is the amount of energy released by the earthquake in joules and $\text{\hspace{0.17em}}{E}_{0}={10}^{4.4}\text{\hspace{0.17em}}$ is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing $\text{\hspace{0.17em}}1.4\cdot {10}^{13}\text{\hspace{0.17em}}$ joules of energy?

about $\text{\hspace{0.17em}}5.83$

## Extensions

Use the definition of a logarithm along with the one-to-one property of logarithms to prove that $\text{\hspace{0.17em}}{b}^{{\mathrm{log}}_{b}x}=x.$

Recall the formula for continually compounding interest, $\text{\hspace{0.17em}}y=A{e}^{kt}.\text{\hspace{0.17em}}$ Use the definition of a logarithm along with properties of logarithms to solve the formula for time $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is equal to a single logarithm.

$t=\mathrm{ln}\left({\left(\frac{y}{A}\right)}^{\frac{1}{k}}\right)$

Recall the compound interest formula $\text{\hspace{0.17em}}A=a{\left(1+\frac{r}{k}\right)}^{kt}.\text{\hspace{0.17em}}$ Use the definition of a logarithm along with properties of logarithms to solve the formula for time $\text{\hspace{0.17em}}t.$

Newton’s Law of Cooling states that the temperature $\text{\hspace{0.17em}}T\text{\hspace{0.17em}}$ of an object at any time t can be described by the equation $\text{\hspace{0.17em}}T={T}_{s}+\left({T}_{0}-{T}_{s}\right){e}^{-kt},$ where $\text{\hspace{0.17em}}{T}_{s}\text{\hspace{0.17em}}$ is the temperature of the surrounding environment, $\text{\hspace{0.17em}}{T}_{0}\text{\hspace{0.17em}}$ is the initial temperature of the object, and $\text{\hspace{0.17em}}k\text{}$ is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is equal to a single logarithm.

$t=\mathrm{ln}\left({\left(\frac{T-{T}_{s}}{{T}_{0}-{T}_{s}}\right)}^{-\text{\hspace{0.17em}}\frac{1}{k}}\right)$

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
Do somebody tell me a best nano engineering book for beginners?
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
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's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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