To determine the solution for the equation
[tex]\[ 2^x = \frac{1}{32}, \][/tex]
we should follow a series of logical steps to find the value of [tex]\( x \)[/tex].
1. Identify the form of the right-hand side:
We know that [tex]\(\frac{1}{32}\)[/tex] is a fraction and we can rewrite it with the base of 2. Notice that [tex]\( 32 \)[/tex] can be written as [tex]\( 2^5 \)[/tex]. Thus,
[tex]\[
\frac{1}{32} = \frac{1}{2^5}.
\][/tex]
2. Use properties of exponents:
Recall that [tex]\(\frac{1}{a^b} = a^{-b}\)[/tex]. Hence,
[tex]\[
\frac{1}{2^5} = 2^{-5}.
\][/tex]
3. Rewrite the original equation:
Now, substitute [tex]\( 2^{-5} \)[/tex] for [tex]\(\frac{1}{32}\)[/tex] in the original equation:
[tex]\[
2^x = 2^{-5}.
\][/tex]
4. Equate the exponents:
Since the bases are the same (both are base 2), we can set the exponents equal to each other:
[tex]\[
x = -5.
\][/tex]
Therefore, the solution to the equation [tex]\( 2^x = \frac{1}{32} \)[/tex] is:
[tex]\[ x = -5. \][/tex]
So the correct choice among the given options is:
```
-5
```
