To determine which function defines [tex]\((g \cdot f)(x)\)[/tex], we need to evaluate the composition of the given functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
Here are the given functions:
[tex]\[
f(x) = \log(5x)
\][/tex]
[tex]\[
g(x) = 5x + 4
\][/tex]
The composition [tex]\((g \cdot f)(x)\)[/tex] means we need to apply [tex]\(f(x)\)[/tex] first and then apply [tex]\(g\)[/tex] to the result of [tex]\(f(x)\)[/tex].
Let's find [tex]\(f(x)\)[/tex] first:
[tex]\[
f(x) = \log(5x)
\][/tex]
Next, we need to find [tex]\(g(f(x))\)[/tex]. This involves substituting [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[
g(f(x)) = g(\log(5x))
\][/tex]
Substitute [tex]\(\log(5x)\)[/tex] into [tex]\(g(y)\)[/tex]:
[tex]\[
g(y) = 5y + 4 \quad \text{where } y = \log(5x)
\][/tex]
Thus,
[tex]\[
g(\log(5x)) = 5 \cdot \log(5x) + 4
\][/tex]
So, the composition function [tex]\((g \cdot f)(x)\)[/tex] is:
[tex]\[
(g \cdot f)(x) = 5 \log(5x) + 4
\][/tex]
None of the provided answer choices exactly match the result of the composition function we derived. Thus, the correct answer is:
[tex]\[
\boxed{\text{None of the given options match the derived composition function.}}
\][/tex]
