Question

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

Answer 1

$\left.e^{2 x}-11 e^{x}-45 e^{-x}+\frac{81}{2}=0\right]$
$\left.\left(e^{x}\right)^{3}-11\left(e^{x}\right)^{2}-45+\frac{81 e^{x}}{2}=0\right]$
$e^{x}=t$
$2 t^{3}-22 t^{2}+81 t-90=0$
$t_{1} t_{2} t_{3}=45$
$e^{x_{1}} \cdot e^{x_{2}} \cdot e^{x_{3}}=45$
$e^{x_{1}+x_{2}+x_{3}}=45$
$\log _{e} e^{x_{1}+x_{2}+x_{3}}=\log _{e} 45$
$x_{1}+x_{2}+x_{3}=\log _{e} 45$
$\log _{e} P=\log _{e} 45$
$P =45$