Question

If $\log _{0.2}(x-1)>\log _{0.04}(x+5)$, then

• A. $-1 < x < 4$
• B. $2 < x < 3$
• C. $1 < x < 4$
• D. $1 < x < 3$

Answer 1

The Correct Option is C

We have, $\log _{0.2}(x-1)>\log _{0.04}(x+5)$ $\Rightarrow \log _{0.2}(x-1)>\log _{(0.2)^{2}}(x+5)$ $\Rightarrow \log _{0.2}(x-1)>\frac{1}{2} \log _{0.2}(x+5)$ $\Rightarrow 2 \log _{0.2}(x-1)>\log _{0.2}(x+5)$ $\Rightarrow \log _{0.2}(x-1)^{2}>\log _{0.2}(x+5)$ $\Rightarrow (x-1)^{2}< x+5$ $\left[\because \log _{a} x >\log _{a} y \Rightarrow x< y\right.$, if $\left.0< a< 1\right]$ $\Rightarrow x^{2}-2 x+1< x+5$ $\Rightarrow x^{2}-3 x-4<0$ $\Rightarrow x^{2}-4 x+x-4< 0$ $\Rightarrow x(x-4)+1(x-4)<0$ $\Rightarrow x(x-4)+1(x-4)<0$ $\Rightarrow (x-4)(x+1)<0$ $\Rightarrow x \in(-1,4)$ But $x>1$ $\Rightarrow x \in(1,4)$