Find the number of solution of `theta in [0,2pi]` satisfying the equation `((log)_(sqrt(3))tantheta(sqrt((log)_(tantheta)3+(log)_(sqrt(3))3sqrt(3)=-1)`

Explanation

`(log_(sqrt(3))tan theta)[sqrt((log_(sqrt(3))3)/(log_(sqrt(3))tan theta)+log_(e) (sqrt(3))^(3))]=-1`
`(log_(sqrt(3))tan theta)[sqrt(2/(log_(sqrt(3))tan theta)+3)]=-1`
Let `log_(sqrt(3))tan theta=y`
`rArr ysqrt(2/y+3)=-1`
`rArr sqrt(2/y+3) = (-1)/y`
`rArr 2/y+3=1/y^(2)" "`(where `y lt 0`)
`rArr y[3y^(2) +2y-1]=0`
`rArr y(3y-1) (y+1)=0`
`rArr y=-1" "( :' y" cannot be positive")`
`rArr log_(sqrt(3))tan theta =-1`
`rArr tan theta=1/sqrt(3)`
`:. theta=pi/6 and (7pi)/6`
Thus, there are two values of `theta` in `[0, 2pi]`