Question

If ${\log }_{2}x+{\log }_{4}x+{\log }_{16}x=\frac{7}{2}$ find the value of $x$.

Answer 1

$\begin{array}{c}{\log }_{2}x+{\log }_{4}x+{\log }_{16}x=\frac{7}{2}\\ \Rightarrow {\log }_{2}x\left[1+\frac{1}{2}+\frac{1}{4}\right]=\frac{7}{2}\\ \Rightarrow {\log }_{2}x\left[\frac{3}{2}+\frac{1}{4}\right]=\frac{7}{2}\\ \Rightarrow {\log }_{2}x\left(\frac{7}{4}\right)=\frac{7}{2}\\ \Rightarrow {\log }_{2}x=2\Rightarrow x=4\end{array}$

Hence, this is the answer.