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Question

The number of solution(s) of the equation (log2cosθ)2+log4cosθ(16cosθ)=2 in the interval [0,2π) is

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Solution

Let cosθ=t
Then, (log2t)2+log4/t(16t)=2
(log2t)2+log2(16t)log2(4t)=2 (Using base change property)
(log2t)2+4+log2t2log2t=2

Let log2t=z
z2+4+z2z=2
z(z22z3)=0
z=0,1,3
t=1,12,8

So, cosθ=12 or cosθ=1
θ=2nπ±π3 or θ=2mπ ; n,mI
Hence, the number of solutions in [0,2π) is 3(θ=π3,5π3,0)

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