We are given the equation:

$\log_5(2x - 5) = 2 \log_5 6 - \log_5 4$

Step 1: Simplify the right-hand side of the equation using logarithmic properties.

First, apply the property $a \log_b c = \log_b(c^a)$ to the term $2 \log_5 6$:

$2 \log_5 6 = \log_5(6^2) = \log_5 36$

Now, the equation becomes:

$\log_5(2x - 5) = \log_5 36 - \log_5 4$

Next, use the property $\log_b a - \log_b c = \log_b \left( \frac{a}{c} \right)$ to combine the logarithms on the right-hand side:

$\log_5(2x - 5) = \log_5 \left( \frac{36}{4} \right)$

Simplifying the fraction:

$\frac{36}{4} = 9$

So, the equation becomes:

$\log_5(2x - 5) = \log_5 9$

Step 2: Set the arguments equal to each other.

Since the logarithms have the same base (5), we can set the arguments equal to each other:

$2x - 5 = 9$

Step 3: Solve for $x$.

Solve the equation:

$2x = 9 + 5 = 14$

$x = \frac{14}{2} = 7$

Final Answer:

$x = 7$

Would you like to see the steps in more detail, or do you have any questions about the solution?

Here are some follow-up questions you might find interesting:

1. What is the base of the logarithm in this problem?
2. How would you solve this if the logarithms had different bases?
3. What happens if you use natural logarithms instead of base 5?
4. How does the property of logarithms simplify the equation in the solution?
5. Can logarithmic equations have multiple solutions? Why or why not?

Tip: Remember that logarithms with the same base can be set equal to each other when their arguments are equal!