Question

Based on this information answer the given queston:
Let $a>0,a\ne 1$ and $x>0$
(i) ${\log }_{a}x>0,a>1\iff x>0,a>1$
(ii)${\log }_{a}x>0,0<a<1\iff 0<x<1,0<a<1$
(iii) ${\log }_{a}x>0,a>1\iff 0<x<1,a>1$
(iv) ${\log }_{a}x<0,0<a<1\iff x>1,0<a<1$

Identify the solution set of $x$ if ${\log }_{1/10}(x^2+x+1)>0$

• A. $(-1,0)$
• B. $(0,1)$
• C. $(1,\infty )$
• D. $(\frac{1}{10},\infty )$

Answer 1

The correct option is A $(-1,0)$

Given ${\log }_{\frac{1}{10}}(x^2+x+1)>0$

$\Rightarrow {\log }_{1/10}(x^2+x+1)>0$

$\Rightarrow 0<(x^2+x+1)<1$

Case-1:

$x^2+x+1>0$

Hence $x\in (-\infty ,\infty )$...........[1]

Case-2:

$x^2+x+1<1$

$x^2+x<0$

$\Rightarrow x\in (-1,0)$..............[2]

From $1$ and $2$:

$x\in (-1,0)$