Solve the logarithmic equation. Express your solutions in exact form only. Please show all of your work.
log3(x+2) + log3(x+8) – 2 = log3(x+3)
Log3x + log3(x+2) = log3 24
log3(x+2)(x+8)-2
To solve the logarithmic equation, we will use the properties of logarithms. Here's the step-by-step solution:
Step 1: Combine the two logarithms on the left side using the product property:
log3[(x+2)(x+8)] - 2 = log3(x+3)
Step 2: Apply the quotient property by subtracting the logarithm on the right side from both sides:
log3[(x+2)(x+8)] - log3(x+3) = 2
Step 3: Apply the quotient property again, but this time by combining the logarithms on the left side:
log3[(x+2)(x+8)/(x+3)] = 2
Step 4: Since the logarithm is base 3, we can rewrite the equation in exponential form:
[(x+2)(x+8)/(x+3)] = 3^2
Step 5: Simplify the right side:
(x+2)(x+8)/(x+3) = 9
Step 6: Multiply both sides of the equation by (x+3) to eliminate the fraction:
(x+2)(x+8) = 9(x+3)
Step 7: Expand the equation:
x^2 + 10x + 16 = 9x + 27
Step 8: Move all terms to one side to form a quadratic equation:
x^2 + 10x + 16 - 9x - 27 = 0
Step 9: Combine like terms:
x^2 + x - 11 = 0
Step 10: Solve the quadratic equation. You can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = 1, and c = -11. Substituting the values into the quadratic formula:
x = (-(1) ± √((1)^2 - 4(1)(-11))) / (2(1))
x = (-1 ± √(1 + 44)) / 2
x = (-1 ± √45) / 2
Step 11: Simplify the square root:
x = (-1 ± √9∙5) / 2
x = (-1 ± 3√5) / 2
Therefore, the solution to the logarithmic equation is:
x = (-1 + 3√5) / 2 or x = (-1 - 3√5) / 2