To determine the reflection point of the point \((2, 200)\) across the line defined by the function \(g(x) = \log_{20} x\), we first need to confirm that the point indeed lies on the line \(f(x) = 20x\).
Calculating \(f(2)\):
\[ f(2) = 20 \times 2 = 40 \]
This means the point \((2, 200)\) does not actually lie on the line \(f(x) = 20x\), it's important to clarify that.
Now let's look at the function \(g(x) = \log_{20} x\). To find a point reflecting across this line that corresponds with the y-coordinate of the point, we need to find the coordinates of \((2, 200)\).
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Find the image of the point on \(g(x)\):
\[ g(2) = \log_{20} 2 \]
This value will help us understand how to reflect across the logarithmic line.
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Calculate the point of intersection with the logarithmic function:
We want to find a point \((a, g(a))\) for which the reflection of \((2, 200)\) across \(g(x)\) would yield some point \((x, y)\) where:
- The midpoint between these two points lies on the line of reflection \(g(x)\).
To compute the reflection, you can also find the following linear equation through the slopes of both functions \(f\) and \(g\).
However, to simplify, we assume a linear reflection and need only find the y-coordinate which can be derived from evaluating \(g(x)\):
\[ g(a) = \log_{20} a \]
For reflection along a proper y-value, we can utilize logarithm transformations, which are a bit more complex.
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Find the true reflection point: The point on the curve where we want our reflection should happen ideally lies equidistant from \(g(x)\), therefore we know this point exists at some logarithmic value.
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Using symmetry, the point is found by the reflection: Regardless of additional complexity, we resolve roughly via \((x, y)\) based upon symmetry about the line of reflection chosen.
Without solving all complexities, your reflection will follow symmetry about \(y = \log_{20}(2)\) to yield the endpoint.
To summarize: The process is thus needed for reflection, and a true accurate determination usually needs to find the point uses numerical or graphical methods or calculating derivatives.
So while we know \((2, 200)\) is not truly a part of the line concerned, the reflection \((x', y')\) sought is defined through calculating intersections of known forms. Please replace desired coordinates based on the scenario.
In a more quantitative approach or deeper geometry in log function computations, it can yield more distinct answer points.