6.6 Exponential and logarithmic equations (Page 3/8)

<< Chapter < Page

College algebra

Page 3 / 8

Chapter >> Page >

Try It

Solve $2^{x} = 3^{x + 1} .$

$x = \frac{\ln 3}{\ln (\frac{2}{3})}$

Got questions? Get instant answers now!

Q&A

Is there any way to solve $2^{x} = 3^{x} ?$

Yes. The solution is $x = 0.$

Equations containing e

One common type of exponential equations are those with base $e .$ This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base $e$ on either side, we can use the natural logarithm to solve it.

How To

Given an equation of the form $y = A e^{k t},$ solve for $t .$

Divide both sides of the equation by $A .$
Apply the natural logarithm of both sides of the equation.
Divide both sides of the equation by $k .$

Solve an equation of the form y = Ae ^kt

Solve $100 = 20 e^{2 t} .$

\begin{array}{l} 100 & = 20 e^{2 t} \\ 5 & = e^{2 t} & Divide by the coefficient of the power . \\ \ln 5 & = 2 t & Take ln of both sides . Use the fact that \ln (x) and e^{x} are inverse functions . \\ t & = \frac{\ln 5}{2} & Divide by the coefficient of t . \end{array}

Got questions? Get instant answers now!

Try It

Solve $3 e^{0.5 t} = 11.$

$t = 2 \ln (\frac{11}{3})$ or $\ln {(\frac{11}{3})}^{2}$

Got questions? Get instant answers now!

Q&A

Does every equation of the form $y = A e^{k t}$ have a solution?

No. There is a solution when $k \neq 0,$ and when $y$ and $A$ are either both 0 or neither 0, and they have the same sign. An example of an equation with this form that has no solution is $2 = −3 e^{t} .$

Solving an equation that can be simplified to the form y = Ae ^kt

Solve $4 e^{2 x} + 5 = 12.$

\begin{array}{l} 4 e^{2 x} + 5 = 12 \\ 4 e^{2 x} = 7 & Combine like terms . \\ e^{2 x} = \frac{7}{4} & Divide by the coefficient of the power . \\ 2 x = \ln (\frac{7}{4}) & Take ln of both sides . \\ x = \frac{1}{2} \ln (\frac{7}{4}) & Solve for x . \end{array}

Got questions? Get instant answers now!

Try It

Solve $3 + e^{2 t} = 7 e^{2 t} .$

$t = \ln (\frac{1}{\sqrt{2}}) = - \frac{1}{2} \ln (2)$

Got questions? Get instant answers now!

Extraneous solutions

Sometimes the methods used to solve an equation introduce an extraneous solution , which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.

Solving exponential functions in quadratic form

Solve $e^{2 x} - e^{x} = 56.$

\begin{array}{l} e^{2 x} - e^{x} & = 56 \\ e^{2 x} - e^{x} - 56 & = 0 & Get one side of the equation equal to zero . \\ (e^{x} + 7) (e^{x} - 8) & = 0 & Factor by the FOIL method . \\ e^{x} + 7 & = 0 or e^{x} - 8 = 0 \begin{matrix}  \end{matrix} & If a product is zero, then one factor must be zero . \\ e^{x} & = - 7 {or e}^{x} = 8 & Isolate the exponentials . \\ e^{x} & = 8 & Reject the equation in which the power equals a negative number . \\ x & = \ln 8 & Solve the equation in which the power equals a positive number . \end{array}

Got questions? Get instant answers now!

Try It

Solve $e^{2 x} = e^{x} + 2.$

$x = \ln 2$

Got questions? Get instant answers now!

Q&A

Does every logarithmic equation have a solution?

No. Keep in mind that we can only apply the logarithm to a positive number. Always check for extraneous solutions.

Using the definition of a logarithm to solve logarithmic equations

We have already seen that every logarithmic equation $\log_{b} (x) = y$ is equivalent to the exponential equation $b^{y} = x .$ We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.

For example, consider the equation $\log_{2} (2) + \log_{2} (3 x - 5) = 3.$ To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for $x :$

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.

Kate Reply

To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.

Muhammed

What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?

Kate Reply

what is the change in momentum of a body?

Eunice Reply

what is a capacitor?

Raymond Reply

Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.

Gautam

A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate

Maria Reply

please solve

Sharon

8m/s²

Aishat

What is Thermodynamics

Muordit

velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.

Mehmet

A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat

Saheed Reply

50 m/s due south east

Someone

which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer

Ramon Reply

I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.

Someone

Scratch that

Someone

temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....

Someone

about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle

Someone

definitely of physics

Haryormhidey Reply

how many start and codon

Esrael Reply

what is field

Felix Reply

physics, biology and chemistry this is my Field

ALIYU

field is a region of space under the influence of some physical properties

Collete

what is ogarnic chemistry

WISDOM Reply

determine the slope giving that 3y+ 2x-14=0

WISDOM

Another formula for Acceleration

Belty Reply

a=v/t. a=f/m a

IHUMA

innocent

Adah

pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?

Nassze Reply

how do lnternal energy measures

Esrael

Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.

JALLAH Reply

No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.

Dlovan

Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?

Robert

like charges repel while unlike charges atttact

Raymond

What is specific heat capacity

Destiny Reply

Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).

AI-Robot

specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin

ROKEEB

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 1

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask

	5 Extraneous solutions By OpenStax Read Online Course
	3 Solving exponential equations using logarithms By OpenStax Read Online Course
	41 Biology 41 Osmotic Regulation and Excretion MCQ By OpenStax Start Quiz
	American Government MCQ By Saylor Foundation Start Quiz
	Measurement Experimentation Lab MCQ By Steve Gibbs Start Quiz
	2 Gastrointestinal Pathophysiology Self-Assessment By Laurence Bailen Start Assessment
	9 Sociology 09 Social Stratification in the US MCQ By OpenStax Start Quiz
©flickr: Luis	Chemistry Final Review By Madison Christian Start Exam
	Microeconomics Practice MCQ By Frank Levy Start Test
	3 Microbiology Final 3 By Madison Christian Start Test
	Power Enigeering types of bearing lubrication By Sam Luong Start Quiz
©flickr: Gareth	Professional Etiquette MCQ By Abby Sharp Start Quiz

6.6 Exponential and logarithmic equations (Page 3/8)

Equations containing e

Solve an equation of the form y = Ae kt

Solving an equation that can be simplified to the form y = Ae kt

Extraneous solutions

Solving exponential functions in quadratic form

Using the definition of a logarithm to solve logarithmic equations

Questions & Answers

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Solve an equation of the form y = Ae ^kt

Solving an equation that can be simplified to the form y = Ae ^kt