Question

If $\log \left({\log }_{\left(1/3\right)}(x^2-\frac{10x}{3}+1)\right)$ is defined, then the interval in which $x$ belongs to is

• A. $(0,\frac{1}{3})$
• B. $[0,\frac{1}{3}]$
• C. $[3,\frac{10}{3})$
• D. $(3,\frac{10}{3})$

Answer 1

Answer: (A), (D)

$(3,\frac{10}{3})$
$(0,\frac{1}{3})$

$x^2-\frac{10x}{3}+1<1$

$\Rightarrow x(3x-10)<0$

$\Rightarrow xϵ(0,\frac{10}{3}).....\left(i\right)$

And $x^2-\frac{10}{3}x+1>0$

roots of $x^2-\frac{10}{3}x+1=\frac{1}{3}and3$

$\Rightarrow xϵ(-\infty ,\frac{1}{3})\cup (3,\infty ).....\left(ii\right)$

From equation (1) and (2), we get

$xϵ\left[\right(0,\frac{1}{3})\cup (3,\frac{10}{3}\left)\right]$