# Chapter 4 - Section 4.3 - Logarithmic Functions - Summary Exercises on Inverse, Exponential, and Logarithmic Functions: 34

$x=\dfrac{3}{2}$

#### 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}

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