Answer
a.) $x = -1$
b.) $x = \frac{1}{1000}$
Work Step by Step
$Use$ $the$ $definition$ $of$ $the$ $logarithmic$ $function$ $to$ $find$ $x:$
a.) $\log_2 (\frac{1}{2}) = x$
b.) $\log_{10} x = -3$
a.) $\log_2 (\frac{1}{2}) = x$
Rewrite $\frac{1}{2}$ as $2^{-1}$ [Note:$2^{-1} = \frac{1}{2^1} = \frac{1}{2}$]
$$\log_2 2^{-1} = x$$
Use the Third Property of Logarithms: $\log_a a^x = x$
$$\log_2 2^{-1} = x \rightarrow -1 = x$$
$$x=-1$$
b.) $\log_{10} x = -3$
Rewrite the logarithmic form to exponential form: $\log_b a = c \rightarrow b^c = a$
$$\log_{10} x = -3 \rightarrow 10^{-3} = x$$
$$x = \frac{1}{10^3}$$
$$x = \frac{1}{10\times10\times10}$$
$$x = \frac{1}{1000}$$