Solve by Factoring x^2(1/x)+x(1- natural log of x)-x- natural log of x = natural log of x

$x^{2} (\frac{1}{x}) + x (1 - \ln (x)) - x - \ln (x) = \ln (x)$

Subtract $\ln (x)$ from both sides of the equation.

$x^{2} (\frac{1}{x}) + x (1 - \ln (x)) - x - \ln (x) - \ln (x) = 0$

Simplify $x^{2} (\frac{1}{x}) + x (1 - \ln (x)) - x - \ln (x) - \ln (x)$ .

Subtract $\ln (x)$ from $- \ln (x)$ .

$x^{2} (\frac{1}{x}) + x (1 - \ln (x)) - x - 2 \ln (x) = 0$

Simplify each term.

Cancel the common factor of $x$ .

Factor $x$ out of $x^{2}$ .

$x \cdot x \frac{1}{x} + x (1 - \ln (x)) - x - 2 \ln (x) = 0$

Cancel the common factor.

$x \cdot x \frac{1}{x} + x (1 - \ln (x)) - x - 2 \ln (x) = 0$

Rewrite the expression.

$x + x (1 - \ln (x)) - x - 2 \ln (x) = 0$

Apply the distributive property.

$x + x \cdot 1 + x (- \ln (x)) - x - 2 \ln (x) = 0$

Multiply $x$ by $1$ .

$x + x + x (- \ln (x)) - x - 2 \ln (x) = 0$

Rewrite using the commutative property of multiplication.

$x + x - x \ln (x) - x - 2 \ln (x) = 0$

Simplify $- 2 \ln (x)$ by moving $2$ inside the logarithm.

$x + x - x \ln (x) - x - \ln (x^{2}) = 0$

Combine the opposite terms in $x + x - x \ln (x) - x - \ln (x^{2})$ .

Subtract $x$ from $x$ .

$x - x \ln (x) + 0 - \ln (x^{2}) = 0$

Add $x - x \ln (x)$ and $0$ .

$x - x \ln (x) - \ln (x^{2}) = 0$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 1.54664235$