Let's solve the two logarithmic expressions step by step:

1. Problem 4: $\log_8 1$

The logarithmic expression $\log_b x$ asks for the exponent to which the base $b$ must be raised to produce $x$. In this case, the problem is asking:

$\log_8 1 = ?$

This means, to what power must 8 be raised to get 1?

The logarithmic rule says that for any base $b$, $\log_b 1 = 0$ because:

$b^0 = 1$

So, $\log_8 1 = 0$.

2. Problem 5: $\log_4 4$

The logarithmic expression $\log_4 4$ asks for the exponent to which the base 4 must be raised to produce 4:

$\log_4 4 = ?$

This is simply:

$4^1 = 4$

So, $\log_4 4 = 1$.

Final Answers:

1. $\log_8 1 = 0$
2. $\log_4 4 = 1$

Would you like further clarification on any of these?

Here are some related questions you might find useful:

1. What is the general rule for logarithms of 1?
2. How can you solve logarithms with different bases?
3. What is the inverse operation of a logarithm?
4. How do fractional exponents relate to logarithms?
5. Can logarithmic properties help simplify complex expressions?

Tip: Remember, $\log_b b = 1$ and $\log_b 1 = 0$ for any base $b$. These are fundamental logarithmic properties.