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.
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.