Question

The value of $b$ satisfying ${\log }_{e}2\cdot {\log }_{b}625={\log }_{10}16\cdot {\log }_{e}10$ is

• A. $b=10$
• B. $b=5$
• C. $b=6$
• D. $b=1$

Answer 1

The correct option is B $b=5$

${\log }_{e}2\cdot {\log }_{b}625={\log }_{10}16\cdot {\log }_{e}10$
$\Rightarrow \left(\frac{\log 2}{\log e}\right){\log }_b{5}^4=\left(\frac{\log 2^4}{\log 10}\right)\left(\frac{\log 10}{\log e}\right),[\because {\log }_{b}a=\frac{\log a}{\log a}]$
$\Rightarrow \left(\log 2\right)4{\log }_{b}5=4\log 2,[\because \log a^x=x\log a]$
$\Rightarrow {\log }_{b}5=1$
$\Rightarrow b=5$
Ans: 5