Answer
$5log_4(\frac{2}{3}) \approx-1.462$
Work Step by Step
Step 1. $4^{0.2x}=\frac{2}{3} \longrightarrow 0.2x=log_4(\frac{2}{3}) \longrightarrow x=5log_4(\frac{2}{3})$
Step 2. $x=5log_4(\frac{2}{3})=\frac{5ln(2/3)}{ln4}\approx-1.462$
Step 3. Check answers, $x=5log_4(\frac{2}{3}) \approx-1.462$ fits the original equation.