Evaluating Logarithms

Let

 

\(y=log_{(4/9)}(27/8)^{(3/4)}\\ y= \frac{log(27/8)^{(3/4)}}{log(4/9)}\\ y= \frac{(3/4)log(27/8)}{log(2/3)^2}\\ y= \frac{(3/4)log(3/2)^3}{log(2/3)^2}\\ y= \frac{(9/4)log(3/2)}{2log(2/3)}\\ y= \frac{9log(3/2)}{8log(2/3)}\\ y= \frac{9log(3/2)}{8log(3/2)^{-1}}\\ y= \frac{9log(3/2)}{-8log(3/2)}\\ y=\frac{-9}{8} \)

OR

 

\(log_{(4/9)}(\frac{27}{8})^{3/4}\\ =log_{(4/9)}(\frac{3}{2})^{9/4}\\ =log_{(4/9)}(\frac{2}{3})^{-9/4}\\ =log_{(4/9)}(\frac{4}{9})^{-9/8}\\ =\frac{-9}{8}\)