Question

Let \(a, b, c\) be real numbers, each greater than 1 , such that \(\frac{2}{3} \log _{b} a+\frac{3}{5} \log _{c} b+\frac{5}{2} \log _{a} c=3\). If the value of \(b\) is 9 , then the value of 'a' must be
(A) \(\sqrt[3]{81}\)
(B) \(\frac{27}{2}\)
(C) 18
(D) 27

Answer 1

Ans: (D)
Hint \(: \frac{2 \ln \mathrm{a}}{3 \ln \mathrm{b}}+\frac{3 \ln \mathrm{b}}{5 \ln \mathrm{c}}+\frac{5 \ln \mathrm{c}}{2 \ln \mathrm{a}}=3\)
$$
\begin{aligned}
&\text { By A.M = G.M } \\
&\begin{array}{l}
\frac{2 \ln \mathrm{a}}{3 \ln \mathrm{b}}=1 \\
\Rightarrow \mathrm{a}^{2}=\mathrm{b}^{3} \Rightarrow \mathrm{a}=\left(3^{6}\right)^{1 / 2}=3^{3} \Rightarrow \mathrm{a}=27
\end{array}
\end{aligned}
$$

Answer 2

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