To find the composition of the functions [tex]\( (g \cdot f)(x) \)[/tex], we need to evaluate [tex]\( g(f(x)) \)[/tex]. Let's go through the steps to do this.
1. Determine [tex]\( f(x) \)[/tex]:
Given:
[tex]\[
f(x) = \log(5x)
\][/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
To find [tex]\( g(f(x)) \)[/tex], substitute [tex]\( f(x) \)[/tex] into the function [tex]\( g \)[/tex]. Recall that:
[tex]\[
g(x) = 5x + 4
\][/tex]
So, replace [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex] with [tex]\( f(x) \)[/tex]:
[tex]\[
g(f(x)) = g(\log(5x))
\][/tex]
3. Evaluate [tex]\( g(f(x)) \)[/tex]:
Substitute [tex]\( \log(5x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(\log(5x)) = 5(\log(5x)) + 4
\][/tex]
4. Simplify the expression:
Putting it all together, we get:
[tex]\[
(g \cdot f)(x) = 5 \log(5x) + 4
\][/tex]
Now, let's compare this result with the provided options:
- Option A:
[tex]\[
5x + 4 + \log(5x)
\][/tex]
This does not match our result.
- Option B:
[tex]\[
5x \log(5x) + 4 \log(5x)
\][/tex]
This does not match our result.
- Option C:
[tex]\[
5 \log(5x) + 4
\][/tex]
This is exactly our result.
- Option D:
[tex]\[
5x - 4 - \log(5x)
\][/tex]
This does not match our result.
Therefore, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
