log²2x-4log2x+3=0

Let's simplify the equation step by step:

First, we can combine the first two terms using the quotient rule of logarithms:

log²2x - 4log2x + 3 = log₂((2x)² / 2⁴) + 3 = log₂(x² / 16) + 3

Now we can use the fact that log₂(a) + log₂(b) = log₂(ab) to combine the terms inside the logarithm:

log₂(x² / 16) + 3 = log₂(x²) - log₂(16) + 3 = 2log₂(x) - 4 + 3 = 2log₂(x) - 1

So our equation becomes:

2log₂(x) - 1 = 0

We can solve for log₂(x) by adding 1 to both sides:

2log₂(x) = 1

log₂(x) = 1/2

Finally, we can solve for x by taking the inverse logarithm of both sides:

x = 2^(log₂(x)) = 2^(1/2) = √2

Therefore, the solution to the equation is x = √2.