Answer
a.) $x = -2$
b.) $x = 32$
Work Step by Step
$Use$ $the$ $definition$ $of$ $the$ $logarithmic$ $function$ $to$ $find$ $x:$
a.) $\log_7 (\frac{1}{49}) = x$
b.) $\log_2 x = 5$
a.) $\log_7 (\frac{1}{49}) = x$
Rewrite $\frac{1}{49}$ as $7^{-2}$ [Note: $7^{-2} = \frac{1}{ 7\times7} = \frac{1}{49}$]
$$\log_7 7^{-2} = x$$
Use the Third Property of Logarithms: $\log_a a^x = x$
$$log_7 7^{-2} = x \rightarrow -2 = x$$
$$x = -2$$
b.) $\log_2 x = 5$
Rewrite the logarithmic form to exponential form: $\log_b a = c \rightarrow b^c = a$
$$\log_2 x = 5 \rightarrow 2^5 = x$$
$$x = 2\times2\times2\times2\times2$$
$$x = 32$$
