# How do you solve log_12x=1/2log_12 9+1/3log_12 27?

Feb 18, 2017

$x = 9$

#### Explanation:

${\log}_{12} x = \frac{1}{2} {\log}_{12} 9 + \frac{1}{3} {\log}_{12} 27$

= ${\log}_{12} \left({9}^{\frac{1}{2}} \times {27}^{\frac{1}{3}}\right)$

Hence $x = {9}^{\frac{1}{2}} \times {27}^{\frac{1}{3}}$

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

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

= ${3}^{1} \times {3}^{1}$

= $3 \times 3 = 9$