Question

The domain of the function
$f\left(x\right)=\sqrt{{\left({\log }_{5}2\right)}^{2x+5}-{\left({\log }_{4}25\right)}^{x-7}}$

Answer 1

$f\left(x\right)=\sqrt{(lo{g}_5^2{)}^{2x+5}-(lo{g}_4^{25}{)}^{x-7}}$

$f\left(x\right)=\sqrt{(lo{g}_5^2{)}^{-2x+5}-(lo{g}_2^5{)}^{x-7}}$

$\sqrt{a^{-2x-5}-a^{x-7}}$

$=\sqrt{\frac{1}{a^{2x}.a^5}-\frac{a^x}{a^7}}$

Now,

$\frac{1}{a^{2x}.a^5}-\frac{a^x}{a^7}\ge 0$

$a^{2(1-x)}-a^x\ge 0$

$a^{2(-1-x)}-a^x\ge 0$

$a^{2(1-x)}\ge a^x$

$a^{2-3x}\ge 1[a^o=1]$

$2-3x\ge 0$

$x\le \frac{2}{3}$

$x\in$($-\alpha ,\frac{2}{3}$]