Answer :
Let's solve the given equation step by step. The equation we need to solve is:
[tex]\[ \log_4(x^2 + 1) = \log_4(-2x) \][/tex]
Step 1: Understanding the properties of logarithms
Since the bases of the logarithms on both sides of the equation are the same, we can equate the arguments of the logarithms:
[tex]\[ x^2 + 1 = -2x \][/tex]
Step 2: Rearrange the equation to standard quadratic form
Let's rewrite the equation as follows:
[tex]\[ x^2 + 1 + 2x = 0 \][/tex]
Step 3: Combine like terms
This simplifies to:
[tex]\[ x^2 + 2x + 1 = 0 \][/tex]
Step 4: Solve the quadratic equation
We can factor this quadratic equation:
[tex]\[ (x + 1)^2 = 0 \][/tex]
This gives us:
[tex]\[ x + 1 = 0 \][/tex]
So,
[tex]\[ x = -1 \][/tex]
Step 5: Verify the solution
We need to verify that the solution [tex]\( x = -1 \)[/tex] does not make the arguments of the logarithms negative or undefined. Let's substitute [tex]\( x = -1 \)[/tex] back into the original arguments:
- For the left side: [tex]\( x^2 + 1 = (-1)^2 + 1 = 1 + 1 = 2 \)[/tex]
- For the right side: [tex]\( -2x = -2(-1) = 2 \)[/tex]
Both arguments are valid and positive numbers for the logarithm function. Therefore, [tex]\( x = -1 \)[/tex] is a valid solution.
Thus, the correct answer is:
[tex]\[ \boxed{x = -1} \][/tex]
The solution to the equation is [tex]\( x = -1 \)[/tex]. Therefore, the correct choice is:
B. [tex]\( x = -1 \)[/tex]
[tex]\[ \log_4(x^2 + 1) = \log_4(-2x) \][/tex]
Step 1: Understanding the properties of logarithms
Since the bases of the logarithms on both sides of the equation are the same, we can equate the arguments of the logarithms:
[tex]\[ x^2 + 1 = -2x \][/tex]
Step 2: Rearrange the equation to standard quadratic form
Let's rewrite the equation as follows:
[tex]\[ x^2 + 1 + 2x = 0 \][/tex]
Step 3: Combine like terms
This simplifies to:
[tex]\[ x^2 + 2x + 1 = 0 \][/tex]
Step 4: Solve the quadratic equation
We can factor this quadratic equation:
[tex]\[ (x + 1)^2 = 0 \][/tex]
This gives us:
[tex]\[ x + 1 = 0 \][/tex]
So,
[tex]\[ x = -1 \][/tex]
Step 5: Verify the solution
We need to verify that the solution [tex]\( x = -1 \)[/tex] does not make the arguments of the logarithms negative or undefined. Let's substitute [tex]\( x = -1 \)[/tex] back into the original arguments:
- For the left side: [tex]\( x^2 + 1 = (-1)^2 + 1 = 1 + 1 = 2 \)[/tex]
- For the right side: [tex]\( -2x = -2(-1) = 2 \)[/tex]
Both arguments are valid and positive numbers for the logarithm function. Therefore, [tex]\( x = -1 \)[/tex] is a valid solution.
Thus, the correct answer is:
[tex]\[ \boxed{x = -1} \][/tex]
The solution to the equation is [tex]\( x = -1 \)[/tex]. Therefore, the correct choice is:
B. [tex]\( x = -1 \)[/tex]
