Let's solve these logarithmic equations by converting them to their exponential forms.

a. $\log_2(x) = 2$

Convert to exponential form: $2^2 = x$ So, $x = 4$

b. $\log_4(x) = -2$

Convert to exponential form: $4^{-2} = x$ Since $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$, we get: $x = \frac{1}{16}$

c. $\log_5(x) = 4$

Convert to exponential form: $5^4 = x$ Since $5^4 = 625$, we get: $x = 625$

Would you like a more detailed explanation of the steps, or do you have any other questions? Here are five related questions that expand on this topic:

1. How do you derive the exponential form from logarithmic equations in general?
2. What is the significance of negative exponents in logarithmic expressions?
3. Can you solve for $x$ when the base of the logarithm is a fraction (e.g., $\log_{\frac{1}{2}}(x) = 3$)?
4. What happens when logarithmic equations involve non-integer solutions?
5. How can logarithmic properties simplify more complex logarithmic expressions?

Tip: Remember, the logarithmic equation $\log_b(x) = y$ always translates to $b^y = x$, making it easier to solve for $x$.