What is x if Ln9+ln4x^2=2?

May 4, 2018

We wish to find $x$ if $\ln 9 + \ln \left(4 {x}^{2}\right) = 2$. We will need to use some log rules here. Specifically, recall that:

$\ln x + \ln y = \ln \left(x y\right)$
${e}^{\ln \left(x\right)} = x$

Using the above two rules, we can quickly come to an answer.

$\ln 9 + \ln \left(4 {x}^{2}\right) = 2$ (Use Rule 1 to combine.)
$\ln \left(9 \cdot 4 {x}^{2}\right) = 2$
$\ln \left(36 {x}^{2}\right) = 2$ (Use Rule 2 and exponentiate both sides.)
$36 {x}^{2} = {e}^{2}$
${x}^{2} = {e}^{2} / \left(36\right) = {\left(\frac{e}{6}\right)}^{2}$
$x = \pm \frac{e}{6}$