Let's solve the equation:

\[
2 \log x = \log 2 + \log(3x - 4)
\]

### Step 1: Apply logarithmic properties
Use the **logarithmic identity**: 

\[
\log a + \log b = \log (a \cdot b)
\]

So, the right-hand side becomes:

\[
\log 2 + \log(3x - 4) = \log [2(3x - 4)]
\]

Now the equation looks like this:

\[
2 \log x = \log [2(3x - 4)]
\]

### Step 2: Simplify the left-hand side
You can apply the **power rule of logarithms**, which states:

\[
a \log b = \log (b^a)
\]

So, \(2 \log x = \log (x^2)\). Now the equation is:

\[
\log (x^2) = \log [2(3x - 4)]
\]

### Step 3: Equate the arguments of the logarithms
Since the logarithms on both sides are equal, you can equate the arguments:

\[
x^2 = 2(3x - 4)
\]

### Step 4: Solve the quadratic equation
Expand the right-hand side:

\[
x^2 = 6x - 8
\]

Now, rearrange the terms to set the equation equal to zero:

\[
x^2 - 6x + 8 = 0
\]

### Step 5: Solve for \(x\)
This is a quadratic equation. We can solve it using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(x^2 - 6x + 8 = 0\), the coefficients are \(a = 1\), \(b = -6\), and \(c = 8\). Plugging these into the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(8)}}{2(1)}
\]

\[
x = \frac{6 \pm \sqrt{36 - 32}}{2}
\]

\[
x = \frac{6 \pm \sqrt{4}}{2}
\]

\[
x = \frac{6 \pm 2}{2}
\]

This gives two possible solutions:

\[
x = \frac{6 + 2}{2} = 4 \quad \text{or} \quad x = \frac{6 - 2}{2} = 2
\]

### Step 6: Check for extraneous solutions
Since logarithms are only defined for positive arguments, we must check whether both solutions satisfy the original equation:

- For \(x = 4\):
  \[
  3(4) - 4 = 8 \quad \text{and} \quad 2 \log 4 = \log 2 + \log 8
  \]
  This is valid.

- For \(x = 2\):
  \[
  3(2) - 4 = 2 \quad \text{and} \quad 2 \log 2 = \log 2 + \log 2
  \]
  This is also valid.

Thus, both solutions \(x = 4\) and \(x = 2\) are valid.

### Final Answer:
\[
x = 4 \quad \text{or} \quad x = 2
\]

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How does the power rule of logarithms work?
2. What is the domain restriction when solving logarithmic equations?
3. How can you apply logarithmic properties to simplify complex expressions?
4. How does the quadratic formula help solve non-linear equations?
5. What are extraneous solutions in the context of logarithmic equations?

**Tip**: Always check the domain when solving logarithmic equations, as the argument of the logarithm must be positive.

This detailed guide walks through solving the equation 2 log x = log 2 + log(3x - 4), applying key logarithmic properties and solving the resulting quadratic equation. Learn how to simplify logarithmic expressions and solve quadratic equations step-by-step. Suitable for Grades 10-12.

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