Chapter 6 - Inverse Functions - 6.3* The Natural Exponential Function - 6.3* Exercises - Page 452: 11

The calculated value for $x$ are ln$3$ and ln$(-2)$.

Given: $e^{2x}-e^{x}-6=0$ Suppose $e^{x} = t$ Then $t^{2}-t-6=0$ gives $t=3,-2$ Thus, $e^{x} = 3,-2$ For $e^{x} = 3$ $x=$ln $3$ For $e^{x} = -2$ $lne^{x}=ln(-2)$ This implies $x=ln(-2)$ Hence, the calculated value for $x$ are ln3 and ln(-2).