To solve the given equation, $\log_2(x+21) = 3$, let's break it down step by step:

1. Rewrite the logarithmic equation in its exponential form: $x + 21 = 2^3$

2. Simplify the exponential expression: $x + 21 = 8$

3. Solve for $x$: $x = 8 - 21$ $x = -13$

4. Verify the solution against the domain of the logarithmic expression: The argument of the logarithmic function, $x + 21$, must be positive: $x + 21 > 0$ $x > -21$ Since $x = -13$ satisfies $x > -21$, the solution is valid.

Final Answer:

The solution set is $\{-13\}$.

Would you like further details or explanations? Here are some related questions for deeper understanding:

1. What are the properties of logarithms used in solving this equation?
2. How do you convert logarithmic equations to exponential form?
3. What happens if the argument of the logarithm is non-positive?
4. Why does $\log_2(x+21)$ require $x > -21$ for the domain?
5. How would this problem change if the base of the logarithm were a different number?

Tip: Always check the domain of logarithmic expressions when solving equations involving them.