Answer
$x=e$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\ln x+\ln 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}
\ln x+2\ln 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)\ln x=3
\\\\
3\ln x=3
\\\\
\ln x=\dfrac{3}{3}
\\\\
\ln x=1
.\end{array}
Since $\ln x=\log_e x,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
\log_e 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}
e^1=x
\\\\
x=e
.\end{array}
