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Chapter 3: Problem 127

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1+\ln x}{2}=0$$

Short Answer

Expert verified
The solution for x in the equation \( \frac{1 + \ln x}{2} = 0 \) is \( x \approx 0.368 \).

Step by step solution

01

Step 1. Isolate the logarithmic term

First, we need to isolate the logarithmic term. Move the constant over to the right side of the equation to isolate the logarithmic term on the left side. Hence, our equation becomes \( \ln x = -1 \).
02

Step 2. Convert the logarithmic equation

Next, we convert the logarithmic equation to an exponential equation. The principle used here is that if \( \ln a = b \), then the equivalent exponential equation is \( e^b = a \). With substitution, our equation becomes \( e^{-1} = x \).
03

Step 3. Perform the calculation

The final step is to solve for x. Once we simplify \( e^{-1} \), we find that \( x \approx 0.368 \) rounded to three decimal places.
04

Step 4. Confirmation through graphical utility

Finally, verifying the solution graphically is done by plotting the equation \( y = \frac{1+\ln x}{2} \). According to the graph, the curve intersects the x-axis at \( x \approx 0.368 \), which confirms our solution.

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