Solve the equation below algebraically. Round your result to three decimal places.
$2 x \ln \left( \frac { 1 } { x } \right) - x = 0$
A) 0.607
B) 0.000
C) -2.009
D) 2.086
E) -1.058

Question

Solve the equation below algebraically. Round your result to three decimal places. 
$2 x \ln \left( \frac { 1 } { x } \right) - x = 0$
A) 0.607
B) 0.000
C) -2.009
D) 2.086
E) -1.058

Quizplus · Accepted Answer

First, we can rearrange the equation to isolate the natural logarithm term:

$$2x\ln\left(\frac{1}{x}\right) = x$$

Then, we can divide both sides by x:

$$2\ln\left(\frac{1}{x}\right) = 1$$

Next, we can simplify the left-hand side by using the logarithmic identity $\ln \frac{1}{x} = -\ln x$:

$$2(-\ln x) = 1$$

$$-2\ln x = \frac{1}{2}$$

Finally, we can divide both sides by -2 to solve for x:

$$\ln x = -\frac{1}{4}$$

$$x = e^{-1/4} \approx 0.607$$

Therefore, the answer is A) 0.607.

Solve the Equation Below Algebraically 2xln⁡(1x)−x=02 x \ln \left( \frac { 1 } { x } \right) - x = 02xln(x1​)−x=0

Solve the Equation Below Algebraically $2 x \ln \left( \frac { 1 } { x } \right) - x = 0$