If `9a^2 + 4b^2 = 18ab,` then `log (3a + 2b) =`

Options

A. `"log" 5 + "log" 3 + "log" a + "log" 5b`
B. `"log" 5 + "log" 3 + "log" 3a + "log" b`
C. `"log" 5 + "log" a + "log"b `
D. none of these

Answer

To solve the equation \(9a^2 + 4b^2 = 18ab\) and find \( \log(3a + 2b) \), we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ 9a^2 + 4b^2 = 18ab \]

### Step 2: Rearranging the equation
We can rearrange this equation to form a perfect square. Notice that:
\[ 9a^2 - 18ab + 4b^2 = 0 \]

### Step 3: Recognize the perfect square
The left-hand side can be recognized as a perfect square:
\[ (3a - 2b)^2 = 0 \]
This implies:
\[ 3a - 2b = 0 \quad \Rightarrow \quad 3a = 2b \quad \Rightarrow \quad \frac{a}{b} = \frac{2}{3} \]

### Step 4: Substitute back to find \(3a + 2b\)
Now, we can express \(3a + 2b\) in terms of \(b\):
\[ 3a + 2b = 3\left(\frac{2}{3}b\right) + 2b = 2b + 2b = 4b \]

### Step 5: Find the logarithm
Now, we can find:
\[ \log(3a + 2b) = \log(4b) \]

### Step 6: Use logarithmic properties
Using the property of logarithms:
\[ \log(4b) = \log(4) + \log(b) \]
Since \(4 = 2^2\), we can further simplify:
\[ \log(4) = 2\log(2) \]
Thus:
\[ \log(3a + 2b) = 2\log(2) + \log(b) \]

### Final Answer
The final expression for \( \log(3a + 2b) \) is:
\[ \log(3a + 2b) = 2\log(2) + \log(b) \]

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