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Chapter 3: Problem 69

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 numerical solution to \(2 \ln (x+3)=3\) to three decimal places, as obtained through graphing and verified algebraically, is \(x = e^{3/2} - 3\). The exact approximation depends on the particular graphing utility used, though.

Step by step solution

01

Transform the Equation

The first step is to isolate the natural logarithm on one side of the equation. Divide every term by 2, to obtain: \(\ln (x+3)= \frac{3}{2}\)
02

Convert Logarithmic Equation to Exponential Equation

The next step is to convert the logarithmic equation into exponential form. The base of natural logarithm is \(e\), so the conversion results in: \(e^{3/2} = x+3\)
03

Solve for x

Subtract 3 from both sides to isolate x: \(x = e^{3/2} - 3\)
04

Graph and Approximate

Using a graphing utility, input y = \(2 \ln (x+3)\) and y = 3. Find the x-coordinate where both graphs intersect to obtain a numerical approximation to three decimal places, this will provide the solution to the equation. It should coincide with the algebraic solution from Step 3.
05

Verification

To verify the obtained result algebraically, substitute the approximated value of x back into the original equation \((2 \ln (x+3)=3)\). If both sides equal, then the obtained result is correct.

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Most popular questions from this chapter

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$6 e^{1-x}=25$$

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