<s>[INST] Given $3^x = 12$ find $(\sqrt{4/3})^\frac{1}{x-2}$ Given $3^x = 12$ find $$\left(\sqrt{\dfrac{4}{3}}\right)^\dfrac{1}{x-2}$$ in simple form.
I've faced this problem, in a high school book, but failed to solve.
I've tried to calculate 


*

*$3^{x-2} = \dfrac{12}{9}$

*$x -2 = \log_3 \dfrac{12}{9}$

*$\dfrac{1}{(x -2)} = \dfrac{ \log 3} { \log (\frac43)}$
However, replacing $\dfrac{1}{(x -2)}$ on the power doesn't help for me. 
$$\left(\dfrac{4}{3}\right)^\dfrac{1}{\dfrac{2 \log 3} { \log (\frac43)} } = \left(\dfrac{4}{3}\right)^\dfrac{\log \left(\frac{4}{3}\right)}{2 \log 3 } = \left(\dfrac{4}{3}\right)^\dfrac{2 \log 2 - \log 3}{2 \log 3 } $$
Any hint?
 [/INST] Let $$y= (\sqrt {4/3})^{\frac {1}{x-2}}$$
We have $$\ln y = \frac {\ln (4/3)}{2(x-2)}$$
Since $$x-2 = \frac {\ln (4/3)}{\ln 3}$$
We get $$\ln y = \ln \sqrt 3 $$ which implies $$ y=\sqrt 3 $$
</s>