(d) log \(\bigg(\frac{2^{n+1}}{3^{n-1}}\bigg)^{\frac{n}{2}}\)
\(\displaystyle\sum_{x=1}^n\) log \(\frac{2^x}{3^{x-1}}\) = log \(\big(\frac{2^1}{3^0}\big)\) + log \(\big(\frac{2^2}{3^1}\big)\) + log \(\big(\frac{2^3}{3^2}\big)\) + ..... + log \(\big(\frac{2^n}{3^{n-1}}\big)\)
= log\(\big(\) \(\frac{2^1}{3^0}\). \(\frac{2^2}{3^1}\).\(\frac{2^3}{3^2}\) . ........ \(\frac{2^n}{3^{n-1}}\big)\)
= log \(\big(\)\(\frac{2^{1+2+3+.....+n}}{3^{1+2+3+.....+(n+1)}}\big)\) = log \(\bigg[\frac{2^{\frac{n(n+1)}{2}}}{3^{\frac{n(n-1)}{2}}}\bigg]\) = log \(\bigg[\frac{2^{n+1}}{3^{n-1}}\bigg]^{\frac{n}{2}}\)