To rewrite \( f(x) = \log_3\left(\frac{x^6}{81}\right) \) using the properties of logarithms, we can use the quotient rule and the power rule of logarithms.
- Quotient Rule: \( \log_b\left(\frac{A}{B}\right) = \log_b(A) - \log_b(B) \)
- Power Rule: \( \log_b(A^n) = n \cdot \log_b(A) \)
Applying the quotient rule to \( f(x) \):
\[ f(x) = \log_3(x^6) - \log_3(81) \]
Now, applying the power rule to \( \log_3(x^6) \):
\[ \log_3(x^6) = 6 \cdot \log_3(x) \]
Next, we need to simplify \( \log_3(81) \). Since \( 81 = 3^4 \):
\[ \log_3(81) = \log_3(3^4) = 4 \cdot \log_3(3) = 4 \]
Now substitute back into the equation for \( f(x) \):
\[ f(x) = 6 \cdot \log_3(x) - 4 \]
Thus, the final result is:
\[ \boxed{f(x) = 6 \cdot \log_3(x) - 4} \]