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Functions — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsFunctions3 Marks
Question
Solve for $\mathrm{x}$ :

$\log _2 x+\log _4 x+\log _{16} x=\frac{21}{4}$

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Answer

$\begin{array}{ll}
& \log 2 x+\log x+\log _{16} x=\frac{21}{4} \\
\therefore \quad & \frac{\log x}{\log 2}+\frac{\log x}{\log 4}+\frac{\log x}{\log 16}=\frac{21}{4} \quad \ldots\left[\log x=\frac{\log x}{\log y}\right] \\
\therefore \quad & \frac{\log x}{\log 2}+\frac{\log x}{\log (2)^2}+\frac{\log x}{\log (2)^4}=\frac{21}{4} \\
\therefore \quad & \frac{\log x}{\log 2}+\frac{\log x}{2 \cdot \log 2}+\frac{\log x}{4 \cdot \log 2}=\frac{21}{4} \\
\therefore \quad & \frac{\log x}{\log 2}\left(1+\frac{1}{2}+\frac{1}{4}\right)=\frac{21}{4} \\
\therefore \quad & \frac{\log x}{\log 2} \times\left(\frac{7}{4}\right)=\frac{21}{4} \\
\therefore \quad & \frac{\log x}{\log 2}=3\\
\therefore \quad & \log x=3 \log 2 \\
\therefore \quad & \log x=\log 2^3 \\
\therefore \quad & x=2^3 \\
\therefore \quad & x=8
\end{array}$

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