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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?

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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!

Solve for the exact value of x: log_5(9x) + 2log_5(3) = 2

Learn how to solve the logarithmic equation log_5(9x) + 2log_5(3) = 2 with a step-by-step solution. This problem uses logarithmic identities and properties to simplify and solve the equation. Ideal for students in Grades 10-12.

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