To solve the logarithmic equation \( \log_5{x^3} - \log{x^2} = 2 \), follow these steps:

### Step 1: Apply the logarithmic power rule
The logarithmic power rule states that \( \log_b{(a^n)} = n \cdot \log_b{a} \). Apply this rule to the terms on the left-hand side:
\[
\log_5{x^3} = 3 \cdot \log_5{x}
\quad \text{and} \quad
\log{x^2} = 2 \cdot \log{x}
\]
So the equation becomes:
\[
3 \cdot \log_5{x} - 2 \cdot \log{x} = 2
\]

### Step 2: Change of base (if needed)
If \( \log{x} \) uses base 10, use the change of base formula \( \log_b{x} = \frac{\log{x}}{\log{b}} \) to convert \( \log_5{x} \) to base 10:
\[
\log_5{x} = \frac{\log{x}}{\log{5}}
\]
Substitute this into the equation:
\[
3 \cdot \frac{\log{x}}{\log{5}} - 2 \cdot \log{x} = 2
\]

### Step 3: Simplify the equation
Factor out \( \log{x} \) from both terms:
\[
\log{x} \left( \frac{3}{\log{5}} - 2 \right) = 2
\]

### Step 4: Solve for \( \log{x} \)
First, simplify the term in parentheses:
\[
\frac{3}{\log{5}} - 2
\]
Now solve for \( \log{x} \). Divide both sides by the simplified factor to isolate \( \log{x} \).

Once you find \( \log{x} \), exponentiate both sides to get the value of \( x \).

Would you like a more detailed walkthrough on the simplifications?

### 5 Related Questions:
1. How does the change of base formula work?
2. What is the logarithmic power rule?
3. What are other methods to solve logarithmic equations?
4. How do you simplify logarithmic expressions?
5. How can you solve equations with different logarithmic bases?

### Tip: Always remember to check the domain of logarithmic equations to avoid negative or zero values inside the log.

This detailed guide walks through solving the logarithmic equation log5x3 - logx2 = 2. Learn key concepts such as the logarithmic power rule and the change of base formula, and follow a step-by-step solution tailored for students in Grades 9-12.

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