To transform the expression [tex]\(\log_6\left(\frac{x^{11}}{x-4}\right)\)[/tex] into a sum and/or difference of logarithms, and to express powers as factors, we will use logarithmic properties.
Specifically, we'll use the following properties of logarithms:
1. [tex]\( \log_b \left(\frac{A}{B}\right) = \log_b(A) - \log_b(B) \)[/tex]
2. [tex]\( \log_b(A^C) = C \cdot \log_b(A) \)[/tex]
Let's apply these properties step-by-step:
### Step 1: Break Down the Logarithm of the Fraction
We begin with:
[tex]\[
\log_6\left(\frac{x^{11}}{x-4}\right)
\][/tex]
Using the first property ([tex]\( \log_b \left(\frac{A}{B}\right) = \log_b(A) - \log_b(B) \)[/tex]), we can rewrite this as:
[tex]\[
\log_6(x^{11}) - \log_6(x-4)
\][/tex]
### Step 2: Simplify the Logarithm of the Power
Next, we'll simplify [tex]\(\log_6(x^{11})\)[/tex] using the property [tex]\( \log_b(A^C) = C \cdot \log_b(A) \)[/tex]:
[tex]\[
\log_6(x^{11}) = 11 \cdot \log_6(x)
\][/tex]
### Step 3: Combine the Results
Now we combine the simplified terms:
[tex]\[
11 \cdot \log_6(x) - \log_6(x-4)
\][/tex]
Thus, the expression [tex]\(\log_6\left(\frac{x^{11}}{x-4}\right)\)[/tex] can be written as:
[tex]\[
11 \cdot \log_6(x) - \log_6(x-4)
\][/tex]
So, the final answer is:
[tex]\[
\boxed{11 \cdot \log_6(x) - \log_6(x-4)}
\][/tex]
