Question

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

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

Answer 1

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

Let ${\log }_{10}x=y$ and ${\log }_{x}10=z$
$thereforeyz=1$
Differentiating both sides w.r.t. $z$, we get,
$\Rightarrow y.1+z\frac{dy}{dz}=0$
$\Rightarrow \frac{dy}{dz}=-\frac{y}{z}=-\frac{{\log }_{10}x}{{\log }_{x}10}=-({\log }_{10}x{)}^2$