To solve the equation \( \log_4{9} > \log_4{x} \), follow these steps:

### Step 1: Apply the property of logarithms
The inequality involves logarithms with the same base, so you can drop the logs and directly compare the arguments. This property can be applied because the logarithmic function is increasing for bases greater than 1 (here, base 4).

\[
\log_4{9} > \log_4{x} \quad \Rightarrow \quad 9 > x
\]

### Step 2: Write the solution
The inequality simplifies to:
\[
x < 9
\]

Thus, the solution is \( x \in (0, 9) \), where \( x \) must be greater than 0 because the logarithm is not defined for non-positive numbers.

Would you like further clarification or more details on any step?

Here are 5 related questions to expand your understanding:
1. How do you solve logarithmic inequalities with different bases?
2. What are the properties of logarithmic functions?
3. How can you rewrite a logarithmic inequality in exponential form?
4. What happens to inequalities when logarithmic functions are decreasing?
5. How do logarithms relate to exponents?

**Tip**: Always remember to check the domain of the logarithmic function when solving equations or inequalities involving logarithms.

Learn how to solve the logarithmic inequality Log4 9 > Log4 x, covering key logarithmic properties and how to simplify inequalities. This problem is ideal for high school students (Grades 9-12), focusing on logarithmic functions and inequality rules.

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