Find the interval of the monotonicity of the function f(x)=`log_(e)((log_(e)x)/(x))`

Answer

Increasing :(1,e)
Decreasing :`(e,oo)`

Explanation

`f(x) =log_(e)((log_(e)x)/(x))`
f(x) is defined if `(log_(e)x)/(x)gt0`
For `x in (0,1), (log_(e)x)/(x)lt0`
For `x in (1,oo)(log_(e)x)/(x)gt0`
For `x in (1,oo), (log_(e)x)/(x)gt0`
`therefore` Domain of f(x) is `(1,oo)`
Now `f'(x) =(1)/(log_(e)x/(x)xx(-1log_(e)x)/(x^(2))`
`therefore f'(x) =(1-log_(e)x)/(xlog_(e)x)`
`f(x) gt 0 if 1-log_(e)xgt0 or xlte`
`f(x) lt 0 if 1-log_(e)xlt0 or xgte`
Thus f(x) increases in `(1,e)` and decrease in `(e,oo)`