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  4. If log 0.2(x-1)> log 0.04(x+5), then

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

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Solution:

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 $x>1$
$\Rightarrow x \in(1,4)$