Question

Use the properties of logarithms to correctly rewrite f(x)=log3(x^6/81)
.(1 point)
f(x)=

Answer 1

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.

1. **Quotient Rule**: \( \log_b\left(\frac{A}{B}\right) = \log_b(A) - \log_b(B) \)
2. **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:

\[
\boxed{f(x) = 6 \cdot \log_3(x) - 4}
\]