Question

If the sum of all the roots of the equation $e^{2x}-11e^x-45e^{-x}+\frac{81}{2}=0$ is ${\log }_{c}P$, then $p$ is equal to______.

Answer 1

Answer: 45

$e^{2x}-11e^x-45e^{-x}+\frac{81}{2}=0]$
${\left(e^x\right)}^3-11{\left(e^x\right)}^2-45+\frac{81e^x}{2}=0]$
$e^x=t$
$2t^3-22t^2+81t-90=0$
$t_1{t}_2{t}_3=45$
$e^{x_1}\cdot e^{x_2}\cdot e^{x_3}=45$
$e^{x_1+x_2+x_3}=45$
${\log }_e{e}^{x_1+x_2+x_3}={\log }_{e}45$
$x_1+x_2+x_3={\log }_{e}45$
${\log }_{e}P={\log }_{e}45$
$P=45$