/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 126 Solve the equation algebraically... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 126

Solve the equation algebraically. Round the 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 \(x = 2.718\), verified through both algebraic solution and graphical representation.

Step by step solution

01

Rearrange the equation to isolate logarithmic term

In this case, multiply by \(x^{2}\) to both sides to cancel out the denominator on the left side of the equation. This then produces the equation: \(1 - \ln x = 0\).
02

Isolate the logarithm

Move the 1 to the opposite side of the equation by subtracting 1 from both sides. The equation becomes \(-\ln x = -1\). This can then be written as \(\ln x = 1\).
03

Convert the logarithmic equation

Now, the equation could be converted to exponent form to solve for \(x\). The equation can be written as \(x = e^{1}\).
04

Compute and round off the value

By using a calculator, the value of \(e^{1}\) is approximately 2.718. Rounding it to three decimal places, it gives \(x = 2.718\).
05

Verify the answer using graphing utility

By substituting the value into the original equation, check if the left hand side equals right hand side, or alternatively, plot the graph of the equation \(\frac{1-\ln x}{x^{2}}\) versus \(x\) and see if the graph indeed crosses x-axis at \(x = 2.718\).

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