We are given the equation:

\[
\log_4(7 - 3x) = \log_4(2x - 6) = 3
\]

This equation can be split into two parts:
1. \(\log_4(7 - 3x) = 3\)
2. \(\log_4(2x - 6) = 3\)

### Step 1: Solve \(\log_4(7 - 3x) = 3\)

The logarithmic equation can be rewritten in exponential form:

\[
7 - 3x = 4^3
\]

Since \(4^3 = 64\), we have:

\[
7 - 3x = 64
\]

Solve for \(x\):

\[
-3x = 64 - 7 = 57
\]
\[
x = \frac{-57}{3} = -19
\]

### Step 2: Solve \(\log_4(2x - 6) = 3\)

Similarly, rewrite the logarithmic equation in exponential form:

\[
2x - 6 = 4^3
\]

Again, \(4^3 = 64\), so:

\[
2x - 6 = 64
\]

Solve for \(x\):

\[
2x = 64 + 6 = 70
\]
\[
x = \frac{70}{2} = 35
\]

### Step 3: Check for consistency

We now have two possible values for \(x\): \(-19\) and \(35\). However, since the logarithm argument must be positive, we need to check whether these values satisfy both logarithmic expressions.

- For \(x = -19\):  
  \(7 - 3(-19) = 7 + 57 = 64\) (positive)  
  \(2(-19) - 6 = -38 - 6 = -44\) (negative, which is invalid for a logarithmic function)

Thus, \(x = -19\) is **not valid**.

- For \(x = 35\):  
  \(7 - 3(35) = 7 - 105 = -98\) (negative, which is invalid for a logarithmic function)  
  \(2(35) - 6 = 70 - 6 = 64\) (positive)

Thus, \(x = 35\) is **also not valid**.

### Conclusion:
There is no valid solution for this equation because both values of \(x\) lead to invalid logarithmic expressions.

Would you like more details or clarifications on any part?

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Here are 5 related questions:
1. How can we check the domain of logarithmic functions?
2. What are the properties of logarithms that are useful in solving equations?
3. Can logarithmic equations have no solutions? If so, when?
4. How do we rewrite logarithmic expressions in exponential form?
5. What are the conditions for the existence of logarithmic functions?

**Tip:** Always check the domain of a logarithmic function before solving to avoid invalid solutions.

Explore how to solve the logarithmic equation Log4(7 - 3x) = Log4(2x - 6) = 3 through step-by-step conversion into exponential form and solving for x. Understand the properties of logarithms and the importance of checking the domain of the function. This solution is suitable for students in Grades 10-12.

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