Question

If $y={\log }^{n}x$ , where ${\log }^n$ means $\log \log \log ...$ (repeated $n$ times), then $x\log x{\log }^{2}x{\log }^{3}x...{\log }^{n-1}x{\log }^{n}x\frac{dy}{dx}$ is equal to:

• A. $\log x$
• B. $x$
• C. $\frac{1}{\log x}$
• D. ${\log }^{n}x$

Answer 1

Answer: (D)

Given that, $y={\log }^{n}x$
$\therefore x\log x{\log }^{2}x{\log }^{3}x...{\log }^{n-1}x{\log }^{n}x\times \frac{dy}{dx}$
$=x\log x{\log }^{2}x{\log }^{3}x...{\log }^{n-1}x{\log }^{n}x\times \frac{1}{x\log x{\log }^{2}x{\log }^{3}x...{\log }^{n-1}x}$
$={\log }^{n}x$