Answer
$x=2$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
x=\sqrt{\log_{1/2} \dfrac{1}{16}}
,$ use the properties of logarithms to find the value of the radicand. Then simplify the radical.
$\bf{\text{Solution Details:}}$
Using exponents, the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=\sqrt{\log_{1/2} \left(\dfrac{1}{2}\right)^4}
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the equation above is equivalent
\begin{array}{l}\require{cancel}
x=\sqrt{4\log_{1/2} \dfrac{1}{2}}
.\end{array}
Since $\log_b b =1,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=\sqrt{4(1)}
\\\\
x=\sqrt{4}
\\\\
x=2
.\end{array}
