To solve the equation $x = \log_4 8$, we need to express $x$ in a way that is more easily solvable.

Step 1: Express the logarithm in terms of a more familiar base

We know that: $\log_a b = \frac{\log_c b}{\log_c a}$ where $c$ is any positive number. A common choice for $c$ is 10 (common logarithm) or $e$ (natural logarithm), but for simplicity, we will use base 2 because both 4 and 8 are powers of 2.

Step 2: Rewrite $\log_4 8$ in terms of base 2

First, express 4 and 8 as powers of 2: $4 = 2^2 \quad \text{and} \quad 8 = 2^3$ Then, substitute these into the logarithm: $\log_4 8 = \log_{2^2} 2^3$ Using the change of base formula, we get: $\log_{2^2} 2^3 = \frac{\log_2 2^3}{\log_2 2^2}$

Step 3: Simplify the expression

We know that: $\log_2 2^3 = 3 \quad \text{and} \quad \log_2 2^2 = 2$ So: $\log_4 8 = \frac{3}{2}$

Final Answer:

$x = \frac{3}{2}$

Would you like more details or have any questions?

Here are five related questions for further exploration:

1. How would you solve $x = \log_5 25$?
2. What is $\log_2 32$?
3. How do you convert between logarithms of different bases?
4. Can you solve $x = \log_4 16$?
5. What is the value of $\log_4 64$?

Tip: When dealing with logarithms, expressing the numbers in terms of the same base often simplifies the calculation significantly.