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Chapter 3: Problem 79

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

Short Answer

Expert verified
The solution to the equation is \(x = e^{-2} 鈮� 0.135\). This has been verified algebraically and graphically.

Step by step solution

01

Isolate ln(x)

To isolate 'ln(x)', subtract 1 from both sides of the equation. This gives \(ln(x) = -2\)
02

Use definition of natural logarithm

The definition of natural logarithm can be used to solve for 'x'. The logarithm base 'e' of a number is the exponent to which 'e' must be raised to equal the number. Therefore, we can rewrite this as \(e^{-2} = x\)
03

Calculating the value of 'x'

To find the value of 'x', calculate the value of \(e^{-2}\) using a calculator. Remember rounding off the number to three decimal places.
04

Verifying the solution

Substitute the calculated value of 'x' back into the original equation to see if the left-hand side equals the right-hand side. Also, using the graphing utility, plot the function \(y = \frac{1+\ln x}{2}\) and verify the point at which the y-coordinate is 0 is the same as the value of 'x' calculated.

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