Log3 (2x +1)log3 (2x+1)=3-log3(x-1)log3(x-1)

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Let's first simplify the given equation:

Using the property of logarithms that log_a(b) * log_a(c) = log_a(b) + log_a(c), we can rewrite the left side of the equation as:

log3(2x + 1)^2 = 3 - log3(x - 1)^2

Now, we can use another property of logarithms that log_a(b)^n = n * log_a(b) to further simplify the equation:

2 log3(2x + 1) = 3 - 2 log3(x - 1)

Now, let's isolate the log terms by moving them to one side of the equation:

2 log3(2x + 1) + 2 log3(x - 1) = 3

Now, we can combine the logarithms using the property log_a(b) + log_a(c) = log_a(b * c):

log3((2x + 1)^2 * (x - 1)^2) = 3

Now, we can simplify the expression inside the logarithm:

(2x + 1)^2 * (x - 1)^2 = 3^3

(2x + 1)^2 * (x - 1)^2 = 27

Now, we can expand the terms (2x + 1)^2 and (x - 1)^2:

(4x^2 + 4x + 1) * (x^2 - 2x + 1) = 27

Expanding further:

4x^4 - 8x^3 + 4x^2 + 4x^2 - 8x + 4 - 27 = 0

4x^4 - 8x^3 + 8x^2 - 8x - 23 = 0

This is the simplified form of the given equation.