Solving a Logarithmic Equation

Solve the logarithmic equation \(2\ln x=\ln 9\).

Solution Each logarithm has the same base, so \[ \begin{array}{rcl@{\quad\qquad}l} 2\ln x &=&\ln 9 \\[4pt] \ln x^{2} &=&\ln 9 & {\hbox{\({\color{#0066A7}{r\log _{a} u=\log _{a}{u}^{r}}}\)}} \\[4pt] x^{2} &=&9 & {\color{#0066A7}{{\hbox{If }\log _{a} u=\log _{a}v, \hbox{ then } u=v.}}} \\[4pt] x &=&3\quad \hbox{or}\quad x=-3 \end{array} \]

We discard the solution \(x=-3\) since \(-3\) is not in the domain of \(f( x) =\ln x\). The solution is 3.