1.

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`

Answer» `log_(sqrt3)tantheta[sqrt(log_tantheta 3+log_sqrt3 3sqrt3)]=-1`
`log_(sqrt3) tantheta[sqrt(2/log_sqrt3 tantheta)+3)]=-1`
`ysqrt(2/y+3)=-1`
`y^2(2/y+3)=1`
`2y+3y^2=1`
`3y^2+2y-1=0`
`3y(y+1)-1(y+1)=0`
`(y+1)(3y-1)=0`
`y=-1`
`log_sqrt3 tantheta=-1`
`tantheta=(sqrt3)^(-1)`
`tantheta=1/sqrt3`
`theta=pi/6,7/6pi`
`y=1/3`
`log_sqrt3 tantheta=1/3`
`tantheta=(sqrt3)^(1/3)`.


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