Answer
a.) $x = 36$
b.) $x = 27$
Work Step by Step
$Use$ $the$ $definition$ $of$ $the$ $logarithmic$ $function$ $to$ $find$ $x:$
a.) $\log_x 6 = \frac{1}{2}$
b.) $\log_x 3 = \frac{1}{3}$
a.) $\log_x 6 = \frac{1}{2}$
Rewrite the logarithmic form to exponential form: $\log_b a = c \rightarrow b^c = a$
$$\log_x 6 = \frac{1}{2} \rightarrow x^{\frac{1}{2}} = 6$$
Rewrite 6 as $36^{\frac{1}{2}}$ [Note: $36^{\frac{1}{2}} = \sqrt 36 = 6$]
$$x^{\frac{1}{2}} = 36^{\frac{1}{2}}$$
$$x = 36$$
b.) $\log_x 3 = \frac{1}{3}$
Rewrite the logarithmic form to exponential form: $\log_b a = c \rightarrow b^c = a$
$$\log_x 3 = \frac{1}{3} \rightarrow x^{\frac{1}{3}} = 3$$
Rewrite 3 as $27^{\frac{1}{3}}$ [Note: $27^{\frac{1}{3}} = \sqrt[3] 27 = 3$]
$$x^{\frac{1}{3}} = 27^{\frac{1}{3}}$$
$$x = 27$$
