## Product of all solutions of the equation ({ln ^2}x + 2ln x – 3 = 0) is equal to – Math Book

Product of all solutions of the equation $${\ln ^2}x + 2\ln x – 3 = 0$$ equal

A. $$\frac{1}{{{e^3}}}.$$

B. $$– 2$$.

C. $$– 3.$$

D. $$\frac{1}{{{e^2}}}.$$

We have: $${\ln ^2}x + 2\ln x – 3 = 0 \Leftrightarrow \left\{ \begin{array}{l}x > 0\\\left( {\ln x – 1} \right) \left( {\ln x + 3} \right)\end{array} \right.\Leftrightarrow \left\{ \begin{array}{l}x > 0\\\left[\begin{array}{l}x=e\\x={e^{–3}}\end{array}\right\end{array}\right\Leftrightarrow\left[\begin{array}{l}x=e\\x={e^{–3}}\end{array}\right$$[\begin{array}{l}x=e\x={e^{–3}}\end{array}\right\end{array}\right\Leftrightarrow\left[\begin{array}{l}x=e\x={e^{–3}}\end{array}\right\)
So $${x_1}. {x_2} = \frac{1}{{{e^2}}}}.$$