1. The problem states that $y = f(x) = \log_m(x)$ with $m > 1$ and asks to find a point that the graph of $y = g(x) = \log_m(0.5x)$ could pass through. 2. Recall the logarithm property: $$\log_m(ab) = \log_m(a) + \log_m(b)$$ 3. Using this property, rewrite $g(x)$: $$g(x) = \log_m(0.5x) = \log_m(0.5) + \log_m(x) = \log_m(0.5) + f(x)$$ 4. This means the graph of $g(x)$ is the graph of $f(x)$ shifted vertically by $\log_m(0.5)$. 5. Since $0.5 < 1$ and $m > 1$, $\log_m(0.5)$ is negative, so the graph shifts downward. 6. Therefore, for any point $(x, y)$ on $f(x)$, the corresponding point on $g(x)$ is $(x, y + \log_m(0.5))$. 7. Given the points: - Point B lies on $f(x)$ near the origin. - Point D is below the x-axis but to the right of B and A. 8. Since $g(x)$ is a downward shift of $f(x)$, the point $g(x)$ passes through corresponding to $x$ of point B will be lower than point B. 9. Point D fits this description as it is below the x-axis and to the right of B. 10. Hence, the graph of $g(x)$ could pass through point D. Final answer: The graph of $g(x)$ could pass through point D.