Question

homework: practice what you missed final exam part a
solve the logarithmic equation.
\log_{6}(x + 3) + \log_{6}(x - 3) = 3
select the correct choice below and, if necessary, fill in the answer box to complete your choice
a. x = \square (simplify your answer. use a comma to separate answers as needed.)
b. there is no solution.

Explanation:

Step1: Apply log addition rule

Using the property \(\log_b M + \log_b N=\log_b(MN)\), we combine the logs:
\(\log_6[(x + 3)(x - 3)] = 3\)

Step2: Convert to exponential form

Recall \(\log_b a = c\) implies \(b^c = a\). So:
\(6^3=(x + 3)(x - 3)\)

Step3: Simplify both sides

Calculate \(6^3 = 216\) and expand the right side: \(x^2 - 9\).
Equation becomes: \(x^2 - 9 = 216\)

Step4: Solve quadratic equation

Add 9 to both sides: \(x^2 = 225\)
Take square roots: \(x=\pm15\)

Step5: Check domain of logs

For \(\log_6(x + 3)\) and \(\log_6(x - 3)\), we need \(x + 3>0\) and \(x - 3>0\), so \(x>3\).

• \(x = 15\): \(15 + 3 = 18>0\), \(15 - 3 = 12>0\) (valid).
• \(x=-15\): \(-15 + 3=-12<0\) (invalid, discard).

Answer:

A. \(x = 15\)