A) \[\sqrt{5}\]
B) 5
C) 25
D) 625
Correct Answer: D
Solution :
[d]We have, |
\[{{\log }_{x}}8=z\]\[\Rightarrow \]\[{{x}^{z}}=8\] |
\[{{\log }_{y}}x=-1\,\Rightarrow x=\frac{1}{y}\Rightarrow xy=1\] |
\[{{\log }_{1/4}}y=-\,1,\]\[y=4,x=\frac{1}{4}\] |
\[\Rightarrow \]\[{{\left( \frac{1}{4} \right)}^{z}}=8\]\[\Rightarrow \]\[{{2}^{-\,2z}}={{2}^{3}}\] |
\[\Rightarrow \]\[z=-\frac{3}{2}\] |
Now, \[{{\left( \frac{1}{x}+1 \right)}^{\log \sqrt{5}({{y}^{2}}+\,4{{z}^{2}})}}\] |
\[{{(5)}^{\log \sqrt{5}\,\left( 16\,+\,4\,\times \,\frac{9}{4} \right)}}\] |
\[={{(\sqrt{5})}^{2\log \sqrt{5}(16\,+\,9)=}}{{\sqrt{5}}^{^{\log \sqrt{5}{{(25)}^{2}}}}}\] |
\[={{25}^{2}}=625\] |
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