Q49 / 155 · WBJEE 2021 · Mathematics

Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be real numbers, each greater than 1 , such that $\frac{2}{3} \log _{\mathrm{b}} \mathrm{a}+\frac{3}{5} \log _{\mathrm{c}} \mathrm{b}+\frac{5}{2} \log _{\mathrm{a}} \mathrm{c}=3$. If the value of $b$ is 9 , then the value of ' $a$ ' must be

  • A
    $\sqrt[3]{81}$
  • B
    $\frac{27}{2}$
  • C
    18
  • Correct answer
    D
    27

Solution

$\frac{2 \ln a}{3 \ln b}+\frac{3 \ln b}{5 \ln c}+\frac{5 \ln c}{2 \ln a}=3$ By A.M $=$ G.M $\frac{2 \ln a}{3 \ln b}=1$ $\Rightarrow a^{2}=b^{3} \Rightarrow a=\left(3^{6}\right)^{1 / 2}=3^{3} \Rightarrow a=27$

Hence, the correct answer is option: 'D'.

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