To analyze the vertical shift of the function \( f(x) = \log_3\left(\frac{x^6}{81}\right) \) compared to its parent function \( f(x) = \log_3(x) \), we can simplify the expression.
- Recall that \( \frac{x^6}{81} = x^6 \cdot \frac{1}{81} = x^6 \cdot 3^{-4} \).
- Rewrite the logarithmic expression using properties of logarithms:
\[ f(x) = \log_3\left(x^6 \cdot 3^{-4}\right) = \log_3(x^6) + \log_3(3^{-4}) = 6 \log_3(x) - 4. \]
From this simplified expression, we see that:
- The term \( 6 \log_3(x) \) represents a vertical stretch by a factor of 6.
- The term \(-4\) represents a vertical shift downward by 4 units.
Thus, compared to the parent function \( f(x) = \log_3(x) \), the function \( f(x) = \log_3\left(\frac{x^6}{81}\right) \) has a vertical shift of 4 units down.
So the correct response is: 4 units down.