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.