Question

The range of $f\left(x\right)={\log }_{5}x+{\log }_{x}5(x>1)$ is

• A. $[0,\infty )$
• B. $(-\infty ,\infty )$
• C. $[2,\infty )$
• D. $[0,2]$

Answer 1

The correct option is C $[2,\infty )$

$f\left(x\right)={\log }_{5}x+{\log }_{x}5(x>1)$

$AM⩾GM$

$\frac{{\log }_{5}x+{\log }_{x}5}{2}⩾\sqrt{{\log }_{5}x\cdot {\log }_{x}5}$

${\log }_{5}x+{\log }_{x}5⩾2\sqrt{\frac{lnx}{ln5}\cdot \frac{ln5}{lnx}}$

${\log }_{5}x+{\log }_{x}5⩾2$

$\therefore$ Range of $f\left(x\right)is[2,\infty )$