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Chapter 3: Problem 115

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$2 \ln (x+3)=3$$

Short Answer

Expert verified
The approximate value of x that solves the logarithmic equation is \(x \approx 1.350\)

Step by step solution

01

Transform the equation

The first step is to transform the logarithmic equation into its equivalent exponential form for easier graphing. Divide both sides of the equation \(2 \ln (x+3)=3\) by 2 to isolate the logarithm on one side. The transformed equation is \(\ln (x+3)=\frac{3}{2}\)
02

Convert logarithmic form to exponential form

Now convert the logarithmic equation \(\ln (x+3)=\frac{3}{2}\) into its equivalent exponential form. The base for natural logarithm (ln) is e, so the equation becomes \(e^{\frac{3}{2}} = x+3\)
03

Solve for x

Subtract 3 from both sides of the equation to find the value of x. This yields \(x= e^{\frac{3}{2}} - 3\)
04

Calculate the approximate value of x

Approximate the value of x to three decimal places using a calculator. This gives \(x \approx 1.350\)
05

Verify the solution

Replace x with the obtained value in the original equation to verify the solution, that is, check if \(2 \ln (1.350 +3) \approx 3\). It does, verifying that \(x \approx 1.350\) is a solution to the equation.

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Most popular questions from this chapter

Determine the time necessary for $$\$ 1000$$to double if it is invested at interest rate \(r\) compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. $$r=6.5 \%$$

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