The equation provided is:

$4^x = 32$

We need to solve for $x$ by expressing both sides as powers of the same base.

1. Rewrite 4 and 32 as powers of 2:
   • $4 = 2^2$
   • $32 = 2^5$

So, we rewrite the equation as:

$(2^2)^x = 2^5$

2. Simplify the left-hand side using the power of a power property $(a^m)^n = a^{mn}$:

$2^{2x} = 2^5$

3. Now, since the bases are the same (both are base 2), we can equate the exponents:

$2x = 5$

4. Solve for $x$:

$x = \frac{5}{2}$

Thus, the solution set is $x = \frac{5}{2}$.

Would you like more details or have any further questions?

Here are five related questions you might explore:

1. How would you solve $3^x = 81$?
2. Can you solve $5^x = 125$ using the same approach?
3. What is the general method for solving exponential equations?
4. How do you solve equations involving logarithms?
5. How can you check your solution for exponential equations?

Tip: When solving exponential equations, always look for ways to express both sides as powers of the same base!