B1 Solve the equation $2 \ln (\sqrt{x})-\ln (1-x)=2$ exactly.
B1. Using the rules for calculating $\mathrm{z}$ logarithms, we transform the given logarithmic equation $\mathrm{v} \ln \left(\frac{x}{1-x}\right)=$ 2. From the definition of the logarithm, it follows that $\frac{x}{1-x}=e^{2}$. We eliminate the fractions and express $x$ from the resulting equation.
Rewriting the equation $\mathrm{v} \ln x-\ln (1-x)=2$ 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-14.jpg?height=74&width=1685&top_left_y=2070&top_left_x=228)

![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-14.jpg?height=68&width=1685&top_left_y=2122&top_left_x=228)

![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-14.jpg?height=63&width=1685&top_left_y=2170&top_left_x=228)

![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-14.jpg?height=60&width=1687&top_left_y=2220&top_left_x=230)

Solution $x=\frac{e^{2}}{1+e^{2}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .1$ point