Question

If $({\log }_{5}x{)}^2+{\log }_{5}x<2$, then $x$ belongs to the interval

• A. $\left(\frac{1}{25},5\right)$
• B. $\left(\frac{1}{5},\frac{1}{\sqrt{5}}\right)$
• C. $(1,\infty )$
• D. $(5,25)$

Answer 1

The correct option is A $\left(\frac{1}{25},5\right)$

Lets assume ${\log }_{5}x=y$
$y^2+y-2<0$
$(y-1)(y+2)<0$
$-2<y<1$
$-2\log 5<\log x<\log 5$
$\log \left(\frac{1}{25}\right)<\log x<log5$
$\frac{1}{25}<x<5$