Question

homework: practice what you missed final exam part a
question
solve the logarithmic equation.
\\( \log_7(x + 3) = 1 - \log_7(x - 3) \\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\( \bigcirc \\) a. \\( x = \square \\) (simplify your answer. use a comma to separate answers as needed.)
\\( \bigcirc \\) b. there is no solution.

Explanation:

Step1: Move log term to left

Add $\log_7(x - 3)$ to both sides: $\log_7(x + 3) + \log_7(x - 3) = 1$

Step2: Apply log product rule

Use $\log_a M + \log_a N = \log_a(MN)$: $\log_7[(x + 3)(x - 3)] = 1$

Step3: Simplify inside log

Expand $(x + 3)(x - 3)$: $\log_7(x^2 - 9) = 1$

Step4: Convert to exponential form

Using $y = \log_a x \Leftrightarrow x = a^y$: $x^2 - 9 = 7^1$

Step5: Solve quadratic equation

Simplify: $x^2 - 9 = 7 \Rightarrow x^2 = 16 \Rightarrow x = \pm 4$

Step6: Check domain

For $\log_7(x + 3)$ and $\log_7(x - 3)$, $x + 3 > 0$ and $x - 3 > 0 \Rightarrow x > 3$. So $x = -4$ is invalid. Check $x = 4$: $4 + 3 = 7 > 0$, $4 - 3 = 1 > 0$, valid.

Answer:

A. $x = 4$