Question

The differential coefficient of $\log_{10} x$ with respect to $\log_{x} 10$ is

• A. $1$
• B. $-(\log_{10} x)^2$
• C. $(\log_{x} 10)^2$
• D. $\frac{x^2}{100}$

Answer 1

The Correct Option is B

Let $u=\log _{10} x$ and $v=\log _{x} 10$
$\Rightarrow u=\frac{\log _{e} x}{\log _{e} 10}$ and
$v=\frac{\log _{e} 10}{\log _{e} x}$
Now, $ \frac{d u}{d x} =\frac{1}{x \log _{e} 10} $
and $ \frac{d v}{d x}=\log _{e} 10\left[\frac{-1}{x\left(\log _{e} x\right)^{2}}\right] $
$\therefore \frac{d u}{d v}=\frac{d u / d x}{d v / d x}=\frac{1}{x \log _{e} 10} \div \frac{-\log _{e} 10}{x\left(\log _{e} x\right)^{2}} $
$=\frac{-\left(\log _{e} x\right)^{2}}{\left(\log _{e} 10\right)^{2}}=-\left(\frac{\log _{e} x}{\log _{e} 10}\right)^{2} $
$=-\left(\log _{10} x\right)^{2} $