To find the point of intersection of the graphs of \( y = \log_5 x \) and \( y = \log_{0.5} x \), we need to set the two equations equal to each other:
\[ \log_5 x = \log_{0.5} x \]
Using the change of base formula for logarithms, we can rewrite both sides:
\[ \frac{\log x}{\log 5} = \frac{\log x}{\log 0.5} \]
Next, we cross-multiply to eliminate the fractions:
\[ \log x \cdot \log 0.5 = \log x \cdot \log 5 \]
Assuming that \( \log x \neq 0 \) (i.e., \( x \neq 1 \)), we can simplify by dividing both sides by \( \log x \):
\[ \log 0.5 = \log 5 \]
This statement is not true, so we consider the case \( \log x = 0 \), which occurs at \( x = 1 \):
\[ \log_5(1) = 0 \quad \text{and} \quad \log_{0.5}(1) = 0 \]
Both equal to 0, implying \( (1, 0) \) is indeed a point of intersection.
Thus, the graphs of \( y = \log_5 x \) and \( y = \log_{0.5} x \) intersect at the point:
\[ \boxed{(1, 0)} \]
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