Answer
$\text{Exact: } \dfrac{5 \ln \left(\dfrac{2}{3} \right)}{ \ln 4}$
$\text{Approximately: } -1.462$
Work Step by Step
Divide both sides of the equation by 0.3:
$$4^{0.2x}= \dfrac{0.2}{0.3} $$
$$4^{0.2x} = \dfrac{2}{3}$$
$\because a^y = b \text{ is equivalent to } y = \log_a b$
$\therefore 4^{0.2x} = \dfrac{2}{3} \text{ is equivalent to }0.2x = \log_4 \left(\dfrac{2}{3} \right)$
SOolve the equation above using the Change of Base Formula, which is $\hspace{20pt} \log_ a M = \dfrac{\log_b M}{\log_b a}$, to obtain:
$0.2x=\log_4 \left(\dfrac{2}{3} \right)\\\\
5(0.2x) = 5\left(\dfrac{\ln \left(\dfrac{2}{3} \right)}{\ln 4}\right)\\\\
x = \dfrac{5\ln \left(\dfrac{2}{3} \right)}{\ln 4}
$
Therefore,
$x =\boxed{ \dfrac{5 \ln \left(\dfrac{2}{3} \right)}{ \ln 4} \approx -1.462}$
