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

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

Desalegn Reply

U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

Jan Reply

this is the study of how the society manages it's scarce resources

Belonwu

what is macroeconomic

John Reply

macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

husaini Reply

what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

Petrus Reply

what is economic.

Milan Reply

scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

Zarshad Reply

Why qualify 28 supplies

Milan

what are explicit costs

Nomsa Reply

out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

David Reply

economic overview notes

Amahle Reply

identify a demand and a supply curve

Salome Reply

i don't know

Parul

there's a difference

Aryan

Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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

	7 Using the one-to-one property of logarithms to solve logarithmic By OpenStax Read Online Course
	5 Extraneous solutions By OpenStax Read Online Course
	11 Clin Sci I - GI Twedt quiz 2 By Brooke Delaney Start Exam
	4 BOD Hemolymphatic -Dr. Han By Brooke Delaney Start Exam
	Assembly Programming Language By JavaChamp Team Start Quiz
	Art History ARTH209 20th Century By Rebecca Butterfield Start Quiz
	Biology 302 Final By Dravida Mahadeo-J... Start Quiz
	Chemistry Final By Briana Hamilton Start Flashcards
	12 Neuroanatomy 12 The Basal Ganglia By Stephen Voron Start Quiz
©flickr: Francisco	U.S. Civil War Pre-test By Danielle Stephens Start Quiz
	3 AP 03 Cellular Level of Organization MCQ By OpenStax Start Quiz
	Interventionism Mixed Economy By Robert Murphy Start Test

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