Solve the logarithmic equation. Please show all of your work.
Loga(x+3) + loga(x-3) = 3
To solve the logarithmic equation Loga(x+3) + loga(x-3) = 3, we can use the following steps:
Step 1: Apply the product rule of logarithms.
According to the product rule, if we have loga(M) + loga(N) = loga(M * N), where M and N are positive numbers.
Applying the product rule to the equation, we can rewrite it as loga((x+3)(x-3)) = 3.
Step 2: Simplify the equation.
Simplifying the right side of the equation: 3 = loga((x+3)(x-3)).
Step 3: Rewrite the equation in exponential form.
In exponential form, loga(M) = N is equivalent to a^N = M.
Rewriting our equation in exponential form, we get a^3 = (x+3)(x-3).
Step 4: Expand and simplify the right side.
Expanding the equation on the right side: a^3 = x^2 - 9.
Step 5: Move all terms to one side of the equation.
Rearranging the equation to have all terms on one side: x^2 - 9 - a^3 = 0.
Step 6: Solve the quadratic equation.
Factoring the quadratic equation: (x - 3)(x + 3) - a^3 = 0.
Now, we have two cases to consider:
Case 1: (x - 3) = 0
Solving for x in this case: x = 3.
Case 2: (x + 3) - a^3 = 0
Solving for x in this case: x = -3 + a^3.
Therefore, the solutions to the logarithmic equation Loga(x+3) + loga(x-3) = 3 are:
1. x = 3
2. x = -3 + a^3.