Answer
a.) $x = \frac{1}{2}$
b.)$x = 16$
Work Step by Step
$Use$ $the$ $definition$ $of$ $the$ $logarithmic$ $function$ $to$ $find$ $x:$
a.) $\log_4 2 = x$
b.) $\log_4 x = 2$
a.) $\log_4 2 = x$
Rewrite 2 as $4^{\frac{1}{2}}$ [Note: $4^{\frac{1}{2}} = \sqrt 4 = 2$]
$$\log_4 4^{\frac{1}{2}} = x$$
Use the Third Property of Logarithms:$\log_a a^x = x$
$$\log_4 4^{\frac{1}{2}} = x \rightarrow \frac{1}{2} = x$$
$$x = \frac{1}{2}$$
b.) $\log_4 x = 2$
Rewrite the logarithmic form to exponential form: $\log_b a = c \rightarrow b^c = a$
$$\log_4 x = 2 \rightarrow 4^2 = x$$
$$x=16$$