To solve the equation $\log_4 2 = 3x - 8$, we'll break it down into simple steps:

Step 1: Understand the logarithmic expression

The equation $\log_4 2$ represents the logarithm of 2 with base 4. This asks, "To what power must 4 be raised to get 2?"

Step 2: Solve the logarithm

Since $4^{\frac{1}{2}} = 2$, we know that: $\log_4 2 = \frac{1}{2}$ So the equation becomes: $\frac{1}{2} = 3x - 8$

Step 3: Solve for $x$

Now, solve the linear equation $\frac{1}{2} = 3x - 8$.

1. Add 8 to both sides to isolate the term with $x$: $\frac{1}{2} + 8 = 3x$ $\frac{1}{2} + \frac{16}{2} = 3x$ $\frac{17}{2} = 3x$

2. Divide both sides by 3 to solve for $x$: $x = \frac{\frac{17}{2}}{3}$ $x = \frac{17}{6}$

Final Answer

The solution is: $x = \frac{17}{6} \approx 2.83$

Would you like any more details or have any questions?

---

Related Questions:

1. What are the properties of logarithms and how can they be used in equations?
2. How do you solve logarithmic equations with different bases?
3. Can logarithms with a base greater than the number ever be negative?
4. How does the change of base formula work in logarithms?
5. What are some real-world applications of logarithms?

Tip: Remember, logarithms are the inverse operations of exponentiation. This can help simplify complex logarithmic problems.