To solve the equation [tex]\(\log_4(x^2 + 1) = \log_4(-2x)\)[/tex], we need to follow a series of steps to simplify and solve the problem.
1. Use the Property of Logarithms:
Since the logs are equal, their arguments must be equal:
[tex]\[
x^2 + 1 = -2x
\][/tex]
2. Rearrange the Equation:
Bring all terms to one side to form a quadratic equation:
[tex]\[
x^2 + 2x + 1 = 0
\][/tex]
3. Factorize the Quadratic:
Notice that the left-hand side is a perfect square trinomial:
[tex]\[
(x + 1)^2 = 0
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Since [tex]\((x + 1)^2 = 0\)[/tex], we have:
[tex]\[
x + 1 = 0 \implies x = -1
\][/tex]
5. Check the Validity of the Solution:
Substitute [tex]\(x = -1\)[/tex] back into the original arguments of the logarithms to ensure it results in real numbers:
- For [tex]\( \log_4(x^2 + 1) \)[/tex]:
[tex]\[
\log_4((-1)^2 + 1) = \log_4(1 + 1) = \log_4(2)
\][/tex]
- For [tex]\( \log_4(-2x) \)[/tex]:
[tex]\[
\log_4(-2(-1)) = \log_4(2)
\][/tex]
Both expressions evaluate to [tex]\(\log_4(2)\)[/tex], indicating that [tex]\(x = -1\)[/tex] is a valid solution.
Therefore, the solution to the equation [tex]\(\log_4(x^2 + 1) = \log_4(-2x)\)[/tex] is:
[tex]\[
\boxed{x = -1}
\][/tex]
Thus, the correct answer is:
A. [tex]\(x = -1\)[/tex]
