Answer
a.) $x = 2$
b.) $x = 4$
Work Step by Step
$Use$ $the$ $definition$ $of$ $the$ $logarithmic$ $function$ $to$ $find$ $x:$
a.) $\log_ x 16 = 4$
b.) $\log_ x 8 = \frac{3}{2}$
a.) $\log_ x 16 = 4$
Rewrite the logarithmic form to exponential form: $\log_b a = c \rightarrow b^c = a$
$$\log_x 16 = 4 \rightarrow x^4 = 16$$
Rewrite 16 as $2^4$ [Note: $2^4 = 2\times2\times2\times2 = 16$]
$$x^4 = 2^4$$
$$x = 2$$
b.) $\log_ x 8 = \frac{3}{2}$
Rewrite the logarithmic form to exponential form: $\log_b a = c \rightarrow b^c = a$
$$\log_x 8 = \frac{3}{2} \rightarrow x^{\frac{3}{2}} = 8$$
$$x^{\frac{3}{2}\times\frac{2}{3}} = 8^{\frac{2}{3}}$$
$$x = \sqrt[3] {64}$$
$$x = 4$$
