tangent through $P(0;3)$ and touch $f(x)=x(1-\ln(x))$ at $P(a;f(a))$

Question

I have problems with this task:

In which point do you have to put the tangents on the graph of $f$ so that this tangent passes through $P (0; 3)$?

Compute the abscissa of $a$.

Answer 1

Hint: Let $$P_1(0;3)$$ and $$P_2(a;f(a))$$ then the slope of our Tangent line is given by : $$m=\frac{y_2-y_1}{x_2-x_1}=\frac{a(1-\ln(a))-3}{a}$$ and this must be

$$f'(a)=-\ln(a)$$ Can you proceed?

Hint: $$a=3$$