Question

Let for all $x>0,f\left(x\right)=\underset{n\to \infty }{lim}n(x^{1/n}-1)$, then

• A. $f\left(x\right)+f\left(\frac{1}{x}\right)=1$
• B. $f\left(xy\right)=f\left(x\right)+f\left(y\right)$
• C. $f\left(xy\right)=xf\left(y\right)+yf\left(x\right)$
• D. $f\left(xy\right)=xf\left(x\right)+yf\left(y\right)$

Answer 1

The correct option is B $f\left(xy\right)=f\left(x\right)+f\left(y\right)$

$f\left(x\right)=\underset{n\to \infty }{lim}n(x^{1/n}-1)=\underset{n\to \infty }{lim}\frac{(x^{1/n}-1)}{1/n}=\log x$

$\log x+\log \frac{1}{x}=0\log xy=\log x+\log y\log xy\ne x\log x+y\log y\ne y\log x+x\log y$