Solve this:
Q.3. If ${\log }_{4}x+{\log }_{2}x=6$, then x must be equal to:
(A) 16
(B) 10
(C) -2
(D) 1
(E) None of these

Dear Student,
$$\begin{gathered} Given : {\log }_{4}x+{\log }_{2}x=6, \\ let \\ {\log }_{4}x=a \\ {\log }_{2}x=b \\ \left[U\sin g Identity, If y = {\log }_b\left(x\right), then x = b^y\right] \\ So, {\log }_{4}x=a can be written as x = 4^a \\ {4}^a=x , 2^b=x \\ {4}^a=2^b \\ \because 2^{2a}=2^b \\ \therefore 2a=b \left(As base are same so we can equate the powers\right) \\ so a+b=6.........\left(i\right) \left[from given equation \right] \\ a+2a=6 \left[b=2a\right] \\ 3a=6 \\ a=\frac{6}{3}=2 \\ As we have \\ {4}^a=x \\ or 2^{2a}=x \\ putting value of a=2 \\ {2}^{2\times 2}=x \\ x=2^4 \\ \mathit{x}\mathbf{=}\mathbf{16} \\ \mathit{H}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathbf{,}\mathbf{ }\mathit{o}\mathit{p}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{ }\left(\mathbf{A}\right)\mathbf{ }\mathit{i}\mathit{s}\mathbf{ }\mathit{c}\mathit{o}\mathit{r}\mathit{r}\mathit{e}\mathit{c}\mathit{t} \end{gathered}$$

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