Answer
$ x\approx 1.10$
Work Step by Step
$ e^{2x}-e^{x}-6=0$
introduce a temporary variable, $ t=e^{x}$ and the equation is equivalent to
$ t^{2}-t-6=0\quad $ ... factor the trinomial
$(t+2)(t-3)=0$
Either $ t=-2$ or $ t=3$
Bring back x:
Either $ e^{x}=-2$ or $ e^{x}=3$
$ e^{x}$ is never negative, so it can't be that $ e^{x}=-2$
This leaves:
$ e^{x}=3\qquad $ ... apply $\ln $( ) to both sides
$ x=\ln 3\qquad $... use calculator and round to two decimal places
$ x\approx 1.10$
