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

Answer: (B)

Let, $u=log_{10} x$ and $v=log_{x} 10$
$\Rightarrow \, u=\frac{log_{e} x}{log_{e} ⁡ 1 0}$ and $v=\frac{log_{e} 1 0}{log_{e} ⁡ x}$
Now, $\frac{d u}{d x}=\frac{1}{x log_{e} 1 0}$
and $\frac{d v}{d x}=\left(log\right)_{e} 10\left[\frac{- 1}{x \left(\left(log\right)_{e} ⁡ x\right)^{2}}\right]$
$\therefore \, \, \frac{d u}{d v}=\frac{d u / d x}{d v / d x}=\frac{1}{x \left(log\right)_{e} 10}\div\frac{- \left(log\right)_{e} ⁡ 1 0}{x \left(\left(log\right)_{e} ⁡ x\right)^{2}}$
$=\frac{- \left(\left(log\right)_{e} x\right)^{2}}{\left(\left(log\right)_{e} ⁡ 10\right)^{2}}=-\left(\frac{\left(log\right)_{e} ⁡ x}{\left(log\right)_{e} ⁡ 10}\right)^{2}$
$=-\left(\left(log\right)_{10} x\right)^{2}$