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The given problem involves logarithmic expressions and a geometric mean \( G \). Let's break it down:

### Given:
- \( \log_9{x} = \log_{36}{y} = a \)
- \( G = \sqrt{xy} \)
- We need to find \( \log_{18}{G} \).

### Step 1: Express \( x \) and \( y \) in terms of \( a \)
From the given information:
- \( \log_9{x} = a \), so \( x = 9^a \)
- \( \log_{36}{y} = a \), so \( y = 36^a \)

### Step 2: Calculate \( G = \sqrt{xy} \)
Now, substitute \( x \) and \( y \):
\[
G = \sqrt{xy} = \sqrt{9^a \cdot 36^a}
\]
This can be rewritten as:
\[
G = \sqrt{(9 \cdot 36)^a} = \sqrt{324^a} = (324^a)^{1/2} = 324^{a/2}
\]
Thus, \( G = 324^{a/2} \).

### Step 3: Calculate \( \log_{18}{G} \)
Now, we need to find \( \log_{18}{G} \). We already have \( G = 324^{a/2} \), so:
\[
\log_{18}{G} = \log_{18}(324^{a/2})
\]
Using the logarithmic property \( \log_b(a^c) = c \log_b(a) \), this becomes:
\[
\log_{18}{G} = \frac{a}{2} \log_{18}{324}
\]

### Step 4: Simplify \( \log_{18}{324} \)
Since \( 324 = 18^2 \), we have:
\[
\log_{18}{324} = \log_{18}{(18^2)} = 2
\]

### Step 5: Final Expression
Substitute \( \log_{18}{324} = 2 \) into the equation:
\[
\log_{18}{G} = \frac{a}{2} \cdot 2 = a
\]

Thus, \( \log_{18}{G} = a \).

### Final Answer: 
\[
\log_{18}{G} = a
\]

Would you like further details or have any other questions?

### Follow-up Questions:
1. How do you simplify logarithmic expressions involving powers and products?
2. What other properties of logarithms can help solve similar problems?
3. How does the base of a logarithm affect its value?
4. Can you think of an example where logarithms of different bases are used?
5. How can logarithmic equations be used to model real-world phenomena?

### Tip:
When dealing with logarithms, always look for opportunities to simplify using logarithmic properties such as the product, quotient, and power rules.

If log9 x = log36 y = a, where x and y are positive real numbers and G = √(xy), then log18 G is equal to

Solve the logarithmic problem where log9 x = log36 y = a, and G = √(xy). Learn how to apply logarithmic rules to simplify and find log18 G step-by-step. This problem covers advanced logarithmic properties and exponent rules, suitable for Grades 10-12.

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