Solve the following loganthmic equation. $$ \log (8 x+9)=1+\log (x-4) $$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (b). The solution set is $$ \} $$. (Simplify your answer. Type an exact answer. Use a comma to separate answers as needed. B. There is no solution.

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To solve the logarithmic equation:

$$
\log (8x + 9) = 1 + \log (x - 4)
$$

**Step 1: Determine the Domain**

For the logarithms to be defined:

$$
8x + 9 > 0 \quad \Rightarrow \quad x > -\frac{9}{8}
$$
$$
x - 4 > 0 \quad \Rightarrow \quad x > 4
$$

Thus, the domain is $$ x > 4 $$.

**Step 2: Solve the Equation**

Rewrite the equation using logarithmic properties:

$$
\log (8x + 9) - \log (x - 4) = 1
$$

Combine the logs:

$$
\log \left( \frac{8x + 9}{x - 4} \right) = 1
$$

Convert from logarithmic to exponential form (assuming base 10):

$$
\frac{8x + 9}{x - 4} = 10^1 = 10
$$

Solve for $$ x $$:

$$
8x + 9 = 10(x - 4)
$$
$$
8x + 9 = 10x - 40
$$
$$
9 + 40 = 10x - 8x
$$
$$
49 = 2x
$$
$$
x = \frac{49}{2}
$$

**Step 3: Verify the Solution**

Ensure that $$ x = \frac{49}{2} $$ is within the domain $$ x > 4 $$:

$$
\frac{49}{2} = 24.5 > 4
$$

**Conclusion**

The solution is valid.

**Answer:**  
The solution set is $$ \left\{\,\dfrac{49}{2}\,\right\} $$.
The solution set is $$ \left\{\,\dfrac{49}{2}\,\right\} $$.

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