To write the given expression as a single logarithm, we will use the properties of logarithms.
Given expression:
[tex]\[ 3 \log_a (9x + 1) + \frac{1}{2} \log_a (x + 7) \][/tex]
### Step 1: Use the Power Rule of Logarithms
The power rule states that [tex]\( k \log_a (b) = \log_a (b^k) \)[/tex]. We'll apply this rule to both terms in the expression.
For the first term:
[tex]\[ 3 \log_a (9x + 1) = \log_a ((9x + 1)^3) \][/tex]
For the second term:
[tex]\[ \frac{1}{2} \log_a (x + 7) = \log_a ((x + 7)^{\frac{1}{2}}) \][/tex]
### Step 2: Combine the Two Logarithms
We now have:
[tex]\[ \log_a ((9x + 1)^3) + \log_a ((x + 7)^{\frac{1}{2}}) \][/tex]
### Step 3: Use the Product Rule of Logarithms
The product rule states that [tex]\( \log_a (b) + \log_a (c) = \log_a (bc) \)[/tex]. We will apply this rule to combine the two logarithms into a single logarithm.
[tex]\[ \log_a ((9x + 1)^3) + \log_a ((x + 7)^{\frac{1}{2}}) = \log_a ((9x + 1)^3 \cdot (x + 7)^{\frac{1}{2}}) \][/tex]
Thus, the expression written as a single logarithm is:
[tex]\[ \log_a ((9x + 1)^3 (x + 7)^{\frac{1}{2}}) \][/tex]
So, the final answer is:
[tex]\[ 3 \log_a (9x + 1) + \frac{1}{2} \log_a (x + 7) = \log_a \left( (9x + 1)^3 (x + 7)^{\frac{1}{2}} \right) \][/tex]
