Question

If ${\log }_{10}x=y$ and ${\log }_{e}10=\frac{1}{m}$, then

• A. ${\log }_{e}x=\frac{y}{m}$
• B. ${\log }_{e}x=\frac{m}{y}$
• C. ${\log }_{e}y=\frac{m}{x}$
• D. ${\log }_{e}y=\frac{x}{m}$
• E. none of these

Answer 1

The correct option is A ${\log }_{e}x=\frac{y}{m}$

Given, ${\log }_{10}x=y$ and ${\log }_{e}10=\frac{1}{m}$
If we multiply those two, we get

$\frac{y}{m}={\log }_{10}x{\log }_{e}10$

$=\frac{\log x}{\log 10}\times \frac{\log 10}{loge}$

$=\frac{\log x}{\log e}={\log }_{e}x$