Let `f(x)=(log)_2(log)_3(log)_4(log)_5(s in x+a^2)dot` Find the set of values of `a` for which the domain of `f(x)i sRdot`

Explanation

Given `f(x)=log_(2)log_(3)log_(4)log_(5)(sinx+a^(2)).`
`f(x)` is defined only if
`log_(3)log_(4)log_(5)(sinx +a^(2)) gt 0 AA x in R`
`or log_(4)log_(5)(sinx +a^(2)) gt 1 AA x in R`
`or log_(5)(sinx +a^(2)) gt 4 AA x in R`
`or (sinx +a^(2)) gt 5^(4) AA x in R`
` or a^(2) gt 625-sinx AA x in R`
Therefore, `a^(2)` must be greater than maximum value of `625 -sinx` which is 626 (when `sinx = -1`). Therefore,
`a^(2) gt 626`
` or a in (-oo,-sqrt(626)) cup (sqrt(626),oo)`