Answer
$-4$
Work Step by Step
Since $16=2^4$, the given expression is equivalent to:
$\log_{\frac{1}{2}}{(2^4)}$
Using the rule $a^{m} = \frac{1}{a^{-m}}, a\ne0$, the expression is equivalent to:
$\log_{\frac{1}{2}}{(2^4)}=\log_{\frac{1}{2}}{\left(\frac{1}{2^{-4}}\right)}$
Note that $\left(\frac{1}{2^{-4}}\right)=\left(\frac{1}{2}\right)^{-4}$.
Thus,
$\log_{\frac{1}{2}}{\left(\frac{1}{2^{-4}}\right)}=\log_{\frac{1}{2}}{\left(\left(\frac{1}{2}\right)^{-4}\right)}$
RECALL:
$\log_a{(a^n)} = n, a \gt 0, a \ne1$
Using the property above gives:
$\log_{\frac{1}{2}}{\left(\left(\frac{1}{2}\right)^{-4}\right)}=-4$
Thus, $\log_{\frac{1}{2}}{16} = -4$.
