https://file.aiquickdraw.com/mathbot/file/172677003666434wb29xp.jpg

The equation shown in the image is:

\[
2 \lg x = \lg (3x - 2)
\]

This involves logarithms (denoted by \(\lg\), which usually represents base-10 logarithms).

### Step-by-step solution:

1. Use the logarithmic property \(a \cdot \lg b = \lg(b^a)\) to simplify the left-hand side:
   \[
   2 \lg x = \lg (x^2)
   \]
   
   So, the equation becomes:
   \[
   \lg (x^2) = \lg (3x - 2)
   \]

2. Now, since the logarithms of both sides are equal, we can set the arguments equal to each other (since \(\lg a = \lg b\) implies \(a = b\)):
   \[
   x^2 = 3x - 2
   \]

3. Rearrange this into a standard quadratic form:
   \[
   x^2 - 3x + 2 = 0
   \]

4. Factor the quadratic equation:
   \[
   (x - 1)(x - 2) = 0
   \]

5. Solve for \(x\):
   \[
   x = 1 \quad \text{or} \quad x = 2
   \]

### Check for extraneous solutions:
Logarithms are only defined for positive arguments. Let's check if both solutions are valid:
- For \(x = 1\):
  \[
  \lg(1) = 0 \quad \text{and} \quad \lg(3(1) - 2) = \lg(1) = 0
  \]
  This solution is valid.
  
- For \(x = 2\):
  \[
  \lg(2) \quad \text{and} \quad \lg(3(2) - 2) = \lg(4)
  \]
  Since both sides give valid logarithms, this solution is also valid.

### Final Answer:
The solutions are \(x = 1\) and \(x = 2\).

Would you like any more details or further clarification on this? Here are five related questions to expand your understanding:

1. What is the property of logarithms that allows us to simplify the left-hand side?
2. Can logarithms of negative numbers exist in the real number system?
3. How would the solution change if a different logarithmic base was used?
4. What are the steps to factor quadratic equations like the one in this solution?
5. How can we determine if a solution is extraneous in logarithmic equations?

**Tip:** Always check the domain when solving logarithmic equations—logarithms are undefined for zero or negative arguments.

Explore the solution to the logarithmic equation 2 lg x = lg(3x - 2). Learn how to simplify logarithmic expressions and solve the resulting quadratic equation, suitable for students in Grades 9-12.

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