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Chapter 5: Q. 41 (page 315)

Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility

31-2x=4x

Short Answer

Expert verified

x=ln32ln3+ln4≈0.307

Step by step solution

01

Step 1. Given informaiton

Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility

31-2x=4x

02

Step 2. Explanation

Firstwenoticethatthebasearenoteaqual.Soweneedtousethistwoproperties.⇒loglaMr=rlogaM⇒IfM=NthanlogaM=logaNForeasiercalculationwewillusethenaturallogarithmlogex=lnx⇒31−2x=4x⇒ln31−2x=ln4x⇒(1−2x)ln3=xln4⇒ln3−2xln3=xln4⇒−2xln3−xln4=−ln3⇒−(2ln3+ln4)=−ln3⇒x=ln32ln3+ln4≈0.307

03

Step 3. Verify the solution with the graph.

The graph of the given function is:

Here we see that the graph is a straight line passes through the point x=0.307.

Thus it is verified that the solution is correct.

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Most popular questions from this chapter

Vehicle Stopping Distance Taking into account reaction time, the distance d(in feet) that a car requires to come to a complete stop while traveling rmiles per hour is given by the function

localid="1646198784356" d(r)=6.97r-90.39

a. Express the speed rat which the car is traveling as a function of the distance localid="1646198792401" drequired to come to a complete stop

b. Verify that localid="1646198797631" r=r(d)is the inverse of localid="1646198802104" d=d(r)by showing that localid="1646198806352" r(d(r))=rand localid="1646198811055" d(r(d))=d

c. Predict the speed that a car was traveling if the distance required to stop was localid="1646198818232" 300feet.

d(r(d))=d

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(b) f(1)

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localid="1646160286433" fx=5-e-x

In Problems 61–72, the function f is one-to-one. Find its inverse and check your answer.

f(x)=42-x

The graph of an exponential function is given. Match each graph to one of the following functions.
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