If a^2=b^3=c^4=d^5, show that loga b c d=47 30 .

Question

If $a^2=b^3=c^4=d^5$, show that loga $b c d=\frac{47}{30}$.

✓

Answer

$
a^2=b^3=c^4=d^5
$
Taking log to the base a throughout, we get
$
\log _a a^2=\log _a b^3=\log _a c^4=\log _a d^5
$
$
\begin{array}{ll}
\therefore & 2 \log _a a=3 \log _a b=4 \log _a c=5 \log _a d \\
\therefore & 2(1)=3 \log _a b=4 \log _a c=5 \log _a d \\
\therefore & \log _a b=\frac{2}{3}, \log _a c=\frac{2}{4}=\frac{1}{2} \text { and } \log _a d=\frac{2}{5} \\
\therefore & \log _a b+\log _a c+\log _a d=\frac{2}{3}+\frac{1}{2}+\frac{2}{5}
\end{array}
$
$\therefore \quad \log _{\mathrm{a}} \mathrm{bcd}=\frac{47}{30}$

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