Let's analyze and solve the given problems:

(a) $4^{\log_4(8)}$

Using the logarithmic property $a^{\log_a(b)} = b$, we have: $4^{\log_4(8)} = 8$

(b) $6^{\log_6(22)}$

Similarly, applying the same property $a^{\log_a(b)} = b$: $6^{\log_6(22)} = 22$

(c) $e^{\ln(30)}$

Using the exponential-logarithmic identity $e^{\ln(a)} = a$: $e^{\ln(30)} = 30$

Final Answers:

1. (a): $8$
2. (b): $22$
3. (c): $30$

Would you like me to explain the logarithmic property used in detail or provide additional examples? 😊

Related Questions:

1. What is the general rule for simplifying $a^{\log_a(b)}$?
2. How do the natural log $\ln$ and base $e$ exponentials interact mathematically?
3. How would the answers change if the logarithms had different bases (e.g., $\log_2$ instead of $\log_4$)?
4. Can $a^{\log_a(b)} = b$ be proven step-by-step?
5. How can properties of logarithms be applied to expressions like $\log_a(xy)$?

Tip:

When simplifying expressions with logarithms and exponents, always check for the identity $a^{\log_a(b)} = b$ and make sure the base matches!