To solve the equation \( \log_4 2 = 3x - 8 \), follow these steps:

### Step 1: Evaluate the logarithm
The logarithm \( \log_4 2 \) asks, "To what power must 4 be raised to get 2?" This can be expressed as:

\[
\log_4 2 = \frac{1}{2}
\]

### Step 2: Substitute the logarithm value into the equation
Now, substitute \( \log_4 2 \) with \( \frac{1}{2} \) in the equation:

\[
\frac{1}{2} = 3x - 8
\]

### Step 3: Solve for \( x \)
Next, solve the equation for \( x \):

1. Add 8 to both sides:
   \[
   \frac{1}{2} + 8 = 3x
   \]

2. Convert 8 to a fraction with a denominator of 2:
   \[
   \frac{1}{2} + \frac{16}{2} = 3x
   \]

3. Add the fractions:
   \[
   \frac{17}{2} = 3x
   \]

4. Divide both sides by 3 to solve for \( x \):
   \[
   x = \frac{17}{6}
   \]

So, the solution is:

\[
x = \frac{17}{6}
\]

Would you like more details on any step or have any questions?

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Here are 5 related questions to consider:
1. How would you solve the equation \( \log_4 8 = 2x - 5 \)?
2. Can you express \( \log_b a \) in terms of common logarithms (base 10)?
3. How would you solve the equation \( \log_2 (x+3) = 4 \)?
4. What is the value of \( x \) in \( \log_{10} x = 3 \)?
5. How do you change the base of a logarithm from base \( b \) to base \( c \)?

**Tip:** When dealing with logarithms, remember that \( \log_b a \) can be rewritten as \( \frac{\log_c a}{\log_c b} \) for any positive base \( c \). This is called the change of base formula.

Learn how to solve the logarithmic equation log base 4 of 2 equals 3x minus 8 with a detailed step-by-step solution. This problem covers key concepts in logarithms and equations, including the change of base formula. Suitable for students in Grades 10-12.

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