The problem asks to solve the logarithmic equation $\log_a(2x+7) = \log_a x + 2\log_a 3$.

1. Problem Description

The problem asks to solve the logarithmic equation $\log_a(2x+7) = \log_a x + 2\log_a 3$.

2. Solution Steps

We are given the equation

$\log_a(2x+7) = \log_a x + 2\log_a 3$

Using the logarithm property $n \log_a x = \log_a x^n$, we can rewrite the equation as:

$\log_a(2x+7) = \log_a x + \log_a 3^2$

$\log_a(2x+7) = \log_a x + \log_a 9$

Using the logarithm property $\log_a x + \log_a y = \log_a (xy)$, we can combine the terms on the right side:

$\log_a(2x+7) = \log_a (9x)$

Since the logarithms have the same base, we can equate the arguments:

$2x+7 = 9x$

Now, we solve for $x$:

$7 = 9x - 2x$

$7 = 7x$

$x = \frac{7}{7}$

$x = 1$

We must check if the solution is valid. We need $2x+7>0$ and $x>0$.

Since $x=1$, we have $2(1)+7=9>0$ and $1>0$.

Therefore, $x=1$ is a valid solution.

3. Final Answer

$x=1$