The line \( f(x) = 20^x \) is an exponential function, and the point \( (2, 200) \) suggests that \( f(2) = 20^2 = 400 \). However, the point given is incorrect because \( 20^2 = 400 \) and not \( 200 \). Therefore, it seems there might be a misunderstanding.
Instead, if we start with the point \( (2, 400) \) that lies on \( f(x) \), we can find the reflection point on the line \( g(x) = \log_{20}(x) \).
To find the reflection, note that \( g(x) \) is the inverse function of \( f(x) \), where: \[ y = f(x) \implies x = g(y) \]
The reflection of a point \( (a, b) \) across the line \( y = x \) is given by \( (b, a) \).
First, let's verify:
- If \( f(2) = 20^2 = 400 \), then the point corresponding to \( g(x) \) would be: \[ g(400) = \log_{20}(400) \]
Using the change of base formula, we can compute \( \log_{20}(400) \): \[ \log_{20}(400) = \frac{\log_{10}(400)}{\log_{10}(20)} \]
Since we can simplify \( 400 = 20^2 \cdot 1 = 20^2 \cdot 20^0 \): \[ 400 = 20^2 \implies \log_{20}(400) = 2 \]
Thus, the reflection point of \( (2, 400) \) across the line \( y = x \) would be: \[ (400, 2) \]
Therefore, the reflection point that lies on \( g(x) = \log_{20}(x) \) is:
\[ \boxed{(400, 2)} \]