Question

# If $$\log _{ 0.3 }{ (x-1) } <\log _{ 0.09 }{ (x-1) }$$, then find the interval in which $$x$$ lies.

A
x>2
B
x<2
C
x>2
D
None of these

Solution

## The correct option is A $$x> 2$$$$log_{0.3}(x-1)<log_{0.09}(x-1)$$ $$\Rightarrow \log_{0.3}(x-1) < \dfrac{\log_{0.3}(x-1)}{\log_{0.3}(0.3)^2}$$$$\Rightarrow \log_{0.3}(x-1) <\dfrac{1}{2} \log_{0.3}(x-1)$$$$\Rightarrow \log_{0.3}(x-1) < \log_{0.3}\sqrt{x-1}$$Taking antilog$$\Rightarrow (x-1)>\sqrt{x-1}$$$$\Rightarrow (x-1)^2-(x-2) >0$$$$\Rightarrow (x-1)(x-2)>0$$ $$As \boxed{x\neq 1}$$$$\Rightarrow \boxed{x>2}$$Mathematics

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