Logarithmic problem- 25 Dec

Question

!!x^(log_5 x) > 5!!

Then !!x in!! ?

!!a)" "(0 , oo)!!

!!b)" "(1/5 , oo)!!

!!c)" "(1/5 , 5)!!

!!d)" "(0 , 1/5) U (5 , oo)!!

Solution

Given equation !!x^(log_5 x) > 5!!

Taking !!log_5!! both sides we get,

!!(log_5 x)(log_5 x) > log_5 5!!

!!=> (log_5 x)^2 - 1 > 0!!

!!=> (log_5 x - 1)((log_5 x + 1) > 0!!

Hence, !!x = 1/5, 5!!

We know that !!x > 0!!,

Hence taking inequality consideration

!!x in (0 , 1/5) U (5 , oo)!!