Question

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

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

Answer 1

The correct option is B $-{\left({\log }_{10}x\right)}^2$

Let $y={\log }_{10}x$ and $z={\log }_{x}10$

We have to find the value of $\frac{dy}{dz}$

Now using base change formula $yz=\frac{\log x}{\log 10}\cdot \frac{\log 10}{\log x}=1$

$\Rightarrow y=\frac{1}{z}$

$\Rightarrow \frac{dy}{dz}=-\frac{1}{z^2}=-y^2=-({\log }_{10}x{)}^2$

Since $\frac{1}{z}=y$