To condense the expression [tex]\(4 \log_9 11 - 4 \log_9 7\)[/tex] to a single logarithm, follow these steps:
1. Factor out the common coefficient:
The expression [tex]\(4 \log_9 11 - 4 \log_9 7\)[/tex] has a common factor of 4. Factor this out:
[tex]\[
4 (\log_9 11 - \log_9 7)
\][/tex]
2. Use the properties of logarithms:
Recall the logarithm property that states [tex]\(\log_b a - \log_b c = \log_b \left( \frac{a}{c} \right)\)[/tex]. Apply this property:
[tex]\[
\log_9 11 - \log_9 7 = \log_9 \left( \frac{11}{7} \right)
\][/tex]
Thus, the expression becomes:
[tex]\[
4 \log_9 \left( \frac{11}{7} \right)
\][/tex]
3. Apply another logarithm property:
Use the power rule for logarithms, which states [tex]\(k \log_b a = \log_b (a^k)\)[/tex]. Apply this rule where [tex]\(k = 4\)[/tex]:
[tex]\[
4 \log_9 \left( \frac{11}{7} \right) = \log_9 \left( \left( \frac{11}{7} \right)^4 \right)
\][/tex]
Therefore, the expression [tex]\(4 \log_9 11 - 4 \log_9 7\)[/tex] condensed to a single logarithm is:
[tex]\[
\log_9 \left( \left( \frac{11}{7} \right)^4 \right)
\][/tex]
This matches the enhanced option:
[tex]\[
\log_9 \left( \left( \frac{11}{7} \right)^4 \right)
\][/tex]
