Question

I $:f\left(x\right)={\log }_{a}x(x>0)$ is an increasing function if $a>1$
II $:f\left(x\right)={\log }_{a}x(x>0)$ is a decreasing function if $0<a<1$
Which of the above statements are true ?

• A. only I
• B. only II
• C. both I and II
• D. neither I nor II

Answer 1

The correct option is C both I and II

(I) $f\left(x\right)=\frac{\log x}{{\log }^a}$
$f^{\prime }\left(x\right)=\frac{1}{x{\log }^a}$
If $f^{\prime }\left(x\right)>0$ than ${\log }^a>0$
Therefore a>1
(II) $f\left(x\right)=\frac{\log x}{{\log }^a}$
$f^{\prime }\left(x\right)=\frac{1}{x{\log }^a}$
If $f^{\prime }\left(x\right)<0$ than ${\log }^a<0$
So $0<a<1$