Let's solve the equation step by step:

We are given the equation:

$\log_5(9x) + 2\log_5(3) = 2$

Step 1: Apply the logarithmic identity

We know that $2\log_b(a) = \log_b(a^2)$. Applying this to the second term, we get:

$\log_5(9x) + \log_5(3^2) = 2$

This simplifies to:

$\log_5(9x) + \log_5(9) = 2$

Step 2: Combine the logarithms

Since we have the sum of two logarithms with the same base, we can use the logarithmic property $\log_b(a) + \log_b(c) = \log_b(a \cdot c)$. Applying this, we get:

$\log_5(9x \cdot 9) = 2$

Simplifying inside the logarithm:

$\log_5(81x) = 2$

Step 3: Remove the logarithm

To remove the logarithm, we rewrite the equation in exponential form. If $\log_b(a) = c$, then $a = b^c$. Applying this to the equation:

$81x = 5^2$

Simplifying:

$81x = 25$

Step 4: Solve for $x$

Now, solve for $x$ by dividing both sides by 81:

$x = \frac{25}{81}$

Thus, the exact value of $x$ is:

$x = \frac{25}{81}$

Would you like more details on any step or have further questions?

Here are five related questions to explore further:

1. What is the general rule for combining logarithms with the same base?
2. How do you convert a logarithmic equation to its exponential form?
3. Can the property $2\log_b(a) = \log_b(a^2)$ be extended to other powers?
4. How would the solution change if the base of the logarithms was different?
5. How can logarithmic identities simplify solving exponential equations?

Tip: Always check whether you can simplify logarithmic expressions by using properties like the product rule, power rule, or change of base formula!