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Chapter 3: Problem 78

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

Short Answer

Expert verified
The solution to the equation is \(x = 2.718\).

Step by step solution

01

Set the Equation to Zero

The given equation is \(\frac{1-\ln x}{x^{2}}=0\). Multiply each side of the equation by \(x^{2}\) to get rid of the denominator. That gives \(1-\ln x = 0\).
02

Solve for x

Rearrange the equation to have \(\ln x\) on one side and constant on the other. This gives \(\ln x = 1\). The base of natural logarithm (ln) is the number 'e'. Therefore, to eliminate the ln, raise 'e' to the power of each side of the equation to solve for x. This gives \(x = e^{1}\).
03

Compute the value of x

Using a calculator, determine the value of \(e^{1}\) and round to three decimal places. The value of e is approximately 2.718, so when raised to the power 1, it remains 2.718.
04

Verify Answer Using Graphing Utility

Graph the function \(\frac{1-\ln x}{x^{2}}\). The function should intersect the x-axis at x = 2.718, further confirming the solution.

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