To condense the given logarithmic expression [tex]\( 6 \log _9 5 + 24 \log _9 11 \)[/tex] into a single logarithm, we can follow a step-by-step process using the properties of logarithms.
### Step 1: Use the Power Rule of Logarithms
The power rule states:
[tex]\[ a \log_b c = \log_b (c^a) \][/tex]
Apply this rule to each term:
[tex]\[ 6 \log _9 5 = \log _9 (5^6) \][/tex]
[tex]\[ 24 \log _9 11 = \log _9 (11^{24}) \][/tex]
### Step 2: Use the Product Rule of Logarithms
The product rule states:
[tex]\[ \log_b x + \log_b y = \log_b (xy) \][/tex]
Combine the two logarithmic terms:
[tex]\[ \log _9 (5^6) + \log _9 (11^{24}) = \log _9 \left( 5^6 \cdot 11^{24} \right) \][/tex]
### Step 3: Simplify the Expression
Now that we have our condensed form, it is:
[tex]\[ \log _9 \left( 5^6 \cdot 11^{24} \right) \][/tex]
### Step 4: Identify the Correct Answer
Comparing this to the provided options, we find that it matches option D:
[tex]\[ \log _9\left(11^{24} \cdot 5^6\right) \][/tex]
Therefore, the answer is:
D) [tex]\(\log _9\left(11^{24} \cdot 5^6\right)\)[/tex]
