Answer
$\text{Exact: } -\dfrac{\ln 1.5}{\ln 2}$
$\text{Approximately: } -0.585$
Work Step by Step
$\because a^y = b \text{ is equivalent to } y = \log_a b$
$\therefore 2^{-x}=1.5 \text{ is equivalent to } -x = \log_2 1.5$
Use the Change of Base Formula, which is $\hspace{20pt} \log_ a M = \dfrac{\log_b M}{\log_b a}$, to obtain:
$-x=\log_2 1.5 \\\\
-x= \dfrac{\ln 1.5}{\ln 2}\\\\
x= -\dfrac{\ln 1.5}{\ln 2}$
Therefore,
$ x = \boxed{-\dfrac{\ln 1.5}{\ln 2} \approx -0.585}$
