How do you solve for x in 3ln3x=6?

Dec 31, 2015

$x = {e}^{2} / 3$

Explanation:

Method One

Divide both sides by $3$.

$\ln 3 x = 2$

To undo the natural logarithm, exponentiate both sides with base $e$.

${e}^{\ln 3 x} = {e}^{2}$

$3 x = {e}^{2}$

$x = \frac{{e}^{2}}{3}$

Method Two

Rewrite the original expression using logarithm rules.

$\ln \left({\left(3 x\right)}^{3}\right) = 6$

$\ln \left(27 {x}^{3}\right) = 6$

${e}^{\ln \left(27 {x}^{3}\right)} = {e}^{6}$

$27 {x}^{3} = {e}^{6}$

${x}^{3} = \frac{{e}^{6}}{27}$

${\left({x}^{3}\right)}^{\frac{1}{3}} = {\left(\frac{{\left({e}^{2}\right)}^{3}}{{3}^{3}}\right)}^{\frac{1}{3}}$

$x = \frac{{e}^{2}}{3}$