If l o g 0.04 ( x − 1 ) ≥ l o g 0.2 ( x − 1 ) then x belongs to the interval Options: A . &nbsp ( 1 , 2 ] B . &nbsp ( − ∞ , 2 ) C . &nbsp [ 2 , + ∞ ] D . &nbsp N o n e o f t h e s e

Answer: Option C : C l o g 0.04 ( x − 1 ) ≥ l o g 0.2 ( x − 1 ) ....(i) For log to be defined x − 1 > 0 ⇒ x > 1 F r o m ( i ) , l o g ( 0.2 ) 2 ( x − 1 ) ≥ l o g 0.2 ( x − 1 ) ⇒ 1 2 l o g 0.2 ( x − 1 ) ≥ l o g 0.2 ( x − 1 ) ⇒ √ x − 1 ≤ ( x − 1 ) ⇒ √ x − 1 ( 1 − √ x − 1 ) ≤ 0 ⇒ 1 − √ x − 1 ≤ 0 ⇒ √ x − 1 ≥ 1 ⇒ x ≥ 2 , ∴ x ϵ [ 2 , ∞ ) .

If log0.04(x−1)≥log0.2(x−1) then x belongs to the interva

Discussion Forum : Inequalities Modulus And Logarithms

Question -

If $l o g_{0.04} (x - 1) \geq l o g_{0.2} (x - 1)$ then x belongs to the interval

Options:

A .

(1, 2]

B .

(- \infty, 2)

C .

[2, + \infty]

D .

N o n e o f t h e s e

Answer: Option C
:
C

l o g_{0.04} (x - 1) \geq l o g_{0.2} (x - 1)

....(i)
For log to be defined

x - 1 > 0 \Rightarrow x > 1

F r o m (i), l o g_{(0.2)^{2}} (x - 1) \geq l o g_{0.2} (x - 1)

\Rightarrow \frac{1}{2} l o g_{0.2} (x - 1) \geq l o g_{0.2} (x - 1) \Rightarrow \sqrt{x - 1} \leq (x - 1) \Rightarrow \sqrt{x - 1} (1 - \sqrt{x - 1}) \leq 0 \Rightarrow 1 - \sqrt{x - 1} \leq 0 \Rightarrow \sqrt{x - 1} \geq 1 \Rightarrow x \geq 2, ∴ x ϵ [2, \infty) .

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Discussion Forum : Inequalities Modulus And Logarithms

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