From the image, the equation provided appears to be:

$\log_2 (3x + 1) = \log_2 (x - 1)$

Let's solve this equation step by step:

Step 1: Use the property of logarithms

If $\log_b (A) = \log_b (B)$, then $A = B$. So we can equate the arguments:

$3x + 1 = x - 1$

Step 2: Solve for $x$

Rearrange the terms:

$3x - x = -1 - 1$

$2x = -2$

$x = -1$

Step 3: Check for validity

For logarithmic functions, the arguments must be positive. Let's check:

1. $3x + 1 > 0$: $3(-1) + 1 = -3 + 1 = -2 \quad (\text{Not valid})$

2. $x - 1 > 0$: $-1 - 1 = -2 \quad (\text{Not valid})$

Since the solution does not satisfy the domain requirements of the logarithmic function, there is no valid solution to this equation.

Would you like additional details or have further questions? Here are related topics to consider:

1. How do the properties of logarithms work in equations?
2. Why do logarithms have domain restrictions?
3. What happens when there are extraneous solutions in logarithmic equations?
4. Can logarithms with different bases be solved similarly?
5. What are the real-world applications of logarithmic equations?

Tip: Always verify the domain of logarithmic arguments after solving equations to avoid invalid solutions.