Q110 / 170 · Manipal (MET) 2017 · Mathematics

$\frac{1}{\log _{25} 10}+\frac{1}{\log _4 10}+\frac{1}{\log _{\sqrt{2}} 10}+\frac{1}{\log _{\sqrt{5}} 10} \text { is }$ equal to

  • A
    $3 / 2$
  • B
    2
  • C
    3
  • Correct answer
    D
    $5 / 2$

Solution

$\begin{aligned} & \frac{1}{\log _{25} 10}+\frac{1}{\log _4 10}+\frac{1}{\log _{\sqrt{2}} 10}+\frac{1}{\log _{\sqrt{5}} 10} \\ = & \frac{\log 25}{\log 10}+\frac{\log \frac{4}{\log 10}}{\log }+\frac{\log \sqrt{2}}{\log 10}+\frac{\log \sqrt{5}}{\log 10} \\ = & \frac{2 \log 5+\frac{1}{2} \log 5}{\log 10}+\frac{2 \log 2+\frac{1}{2} \log 2}{\log 10} \\ = & \frac{5}{2}\left(\frac{\log 5+\log 2}{\log 10}\right)=\frac{5}{2} \frac{\log 10}{\log 10}=\frac{5}{2}\end{aligned}$

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

Relevant Questions