https://file.aiquickdraw.com/mathbot/file/1737566029647pcyhgs8j.jpg

Let's solve the given equation step by step:

### Problem:  
\[
\log_3(2x + 1) - \log_3(x) = 2
\]

### Step 1: Use the logarithmic property  
\[
\log_a(m) - \log_a(n) = \log_a\left(\frac{m}{n}\right)
\]  
So the equation becomes:  
\[
\log_3\left(\frac{2x + 1}{x}\right) = 2
\]

### Step 2: Convert the logarithmic equation to exponential form  
\[
\log_a(b) = c \quad \implies \quad b = a^c
\]  
Here:  
\[
\frac{2x + 1}{x} = 3^2
\]
\[
\frac{2x + 1}{x} = 9
\]

### Step 3: Solve for \(x\)  
Multiply through by \(x\) (assuming \(x \neq 0\)):  
\[
2x + 1 = 9x
\]
Simplify:  
\[
1 = 7x
\]
\[
x = \frac{1}{7}
\]

### Final Answer:  
\[
\boxed{x = \frac{1}{7}}
\]

Would you like a detailed explanation of any specific step? Let me know!  

---

### Related Questions:
1. What are the general properties of logarithms used in solving equations?  
2. How do you handle cases where the logarithmic argument becomes negative or zero?  
3. Can you derive a formula for solving \(\log_a(mx + b) - \log_a(nx + c) = d\)?  
4. How do exponential equations relate to logarithms?  
5. What are the restrictions for the domain of logarithmic functions?

### Tip:  
Always ensure the argument of a logarithm (e.g., \(2x + 1\) and \(x\)) is positive before solving the equation.

Solve the following equation: log3(2x + 1) - log3(x) = 2

Learn how to solve the logarithmic equation log3(2x + 1) - log3(x) = 2 step by step. This problem involves key algebraic concepts like logarithmic properties and exponential relationships, suitable for students in Grades 9-11.

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