Final answer:
To solve the equation log₄(x+12) + log₄(x+6) = 2, we can use the property that the sum of logarithms with the same base is equal to the logarithm of their product. The values of x that satisfy the equation are -14 and -4.
Step-by-step explanation:
To solve the equation log₄(x+12) + log₄(x+6) = 2, we can use the property that the sum of logarithms with the same base is equal to the logarithm of their product. In this case, we have:
log₄((x+12)(x+6)) = 2
Next, we can rewrite the equation in exponential form to get:
4² = (x+12)(x+6)
Simplifying further, we get:
16 = x² + 18x + 72
Now, rearrange the equation:
x² + 18x + 56 = 0
This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, factoring is the easiest method:
(x + 14)(x + 4) = 0
From here, we can set each factor equal to zero and solve for x:
x + 14 = 0 or x + 4 = 0
x = -14 or x = -4
Therefore, the values of x that satisfy the equation are -14 and -4.