# Chapter 4 - Section 4.5 - Exponential and Logarithmic Equations - 4.5 Exercises: 46

$x=10$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given equation, $\log x+\log x^2=3 ,$ use the properties of logarithms to simplify the given expression. Then convert to exponential form. $\bf{\text{Solution Details:}}$ 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} \log x+2\log x=3 .\end{array} Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to \begin{array}{l}\require{cancel} (1+2)\log x=3 \\\\ 3\log x=3 \\\\ \log x=\dfrac{3}{3} \\\\ \log x=1 .\end{array} Since $\log x=\log_{10} x,$ the equation above is equivalent to \begin{array}{l}\require{cancel} \log_{10} x=1 .\end{array} Since $y=b^x$ is equivalent to $\log_b y=x,$ the exponential form of the equation above is \begin{array}{l}\require{cancel} 10^1=x \\\\ x=10 .\end{array}

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