To determine which point lies on the graph of the logarithmic function \( g(x) = \log_9 x \), we can use the relationship between exponential and logarithmic functions.
Given the point \((2, 81)\) lies on the graph of the exponential function \( f(x) = 9^x \), this means:
\[ f(2) = 9^2 = 81. \]
For logarithms, if \( g(x) = \log_9 x \), then the point \((a, b)\) on the logarithmic function corresponds to the exponential form:
\[ g(b) = a \implies 9^a = b. \]
In our case, we can find \( g(81) \):
We want to find \( g(81) = \log_9 81 \).
Since \( 81 = 9^2 \), we can write:
\[ \log_9 81 = 2. \]
Thus, the point \((81, 2)\) lies on the graph of the logarithmic function \( g(x) = \log_9 x \).
Therefore, the correct answer is:
(81, 2).