Q72 / 160 · WBJEE 2015 · Mathematics

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

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

Solution

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

Hence, the correct answer is option: 'C'.

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