Answer
$x=\dfrac{3}{2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
x=\log_2\sqrt{8}
,$ use the definition of rational exponents and the properties of logarithms.
$\bf{\text{Solution Details:}}$
Using exponents, the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=\log_2\sqrt{2^3}
.\end{array}
Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=\log_2 2^{\frac{3}{2}}
.\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=\dfrac{3}{2}\log_2 2
.\end{array}
Since $\log_b b=1,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=\dfrac{3}{2}(1)
\\\\
x=\dfrac{3}{2}
.\end{array}