College Algebra (11th Edition)

$x=2$
$\bf{\text{Solution Outline:}}$ 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}