We are given the equation:

$\log_4 z = \frac{1}{2}$

This means that we are looking for the value of $z$ such that:

$4^{\frac{1}{2}} = z$

We know that $4^{\frac{1}{2}}$ is the square root of 4:

$4^{\frac{1}{2}} = \sqrt{4} = 2$

Therefore, the value of $z$ is:

$z = 2$

So, the solution is $z = 2$.

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Would you like a more detailed explanation, or do you have any questions?

Here are some related questions to explore:

1. What is the general form of solving a logarithmic equation?
2. How do logarithmic functions with different bases behave?
3. What happens when the logarithm equation has a base greater than 1 versus less than 1?
4. How do we solve logarithmic equations involving exponents or powers?
5. What is the inverse of the logarithmic function?

Tip: Always remember that logarithms are exponents in disguise! The equation $\log_b a = c$ is equivalent to $b^c = a$.