Question

If $x\in R$, then the range of ${\log }_{5/4}(5x^2-8x+4)$ is

• A. $(0,\infty )$
• B. $[0,\infty )$
• C. $[-1,\infty )$
• D. $[1,\infty )$

Answer 1

The correct option is C $[-1,\infty )$

$5x^2-8x+4$
$=5(x^2-\frac{8x}{5}+\frac{4}{5})$
$=5{(x-\frac{4}{5})}^2+\frac{4}{5}$
$\therefore \frac{4}{5}\le 5x^2-8x+4<\infty$

As ${\log }_{a}x$ is strictly increasing for $a>1$,
${\log }_{5/4}\left(\frac{4}{5}\right)\le {\log }_{5/4}(5x^2-8x+4)<\infty$
$\Rightarrow -1\le {\log }_{5/4}(5x^2-8x+4)<\infty$

Hence, $R\left(f\right)=[-1,\infty )$