if `a^2+4b^2=12ab,` then `log(a+2b)`

Options

A. `(1)/(2)("log"a + "log"b - "log"2)`
B. `"log"(a)/(2) + "log" (b)/(2) + "log" 2`
C. `(1)/(2)("log"a + "log"b + 4 "log"2)`
D. `(1)/(2)("log"a - "log"b + 4"log" 2)`

Answer

To solve the equation \( a^2 + 4b^2 = 12ab \) and find \( \log(a + 2b) \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ a^2 + 4b^2 = 12ab \]

### Step 2: Rearrange the equation
We can rearrange this equation to isolate the terms:
\[ a^2 - 12ab + 4b^2 = 0 \]

### Step 3: Recognize it as a quadratic equation
This is a quadratic equation in terms of \( a \):
\[ a^2 - 12ab + 4b^2 = 0 \]

### Step 4: Factor the quadratic
We can factor the quadratic equation:
\[ (a - 6b)^2 = 0 \]
This implies:
\[ a - 6b = 0 \quad \Rightarrow \quad a = 6b \]

### Step 5: Substitute \( a \) back into \( a + 2b \)
Now we substitute \( a \) back into the expression \( a + 2b \):
\[ a + 2b = 6b + 2b = 8b \]

### Step 6: Find \( \log(a + 2b) \)
Now we need to find \( \log(a + 2b) \):
\[ \log(a + 2b) = \log(8b) \]

### Step 7: Use logarithmic properties
Using the properties of logarithms, we can write:
\[ \log(8b) = \log(8) + \log(b) \]

### Step 8: Simplify \( \log(8) \)
Since \( 8 = 2^3 \), we have:
\[ \log(8) = \log(2^3) = 3 \log(2) \]

### Step 9: Final expression
Thus, we can express \( \log(a + 2b) \) as:
\[ \log(a + 2b) = 3 \log(2) + \log(b) \]

### Conclusion
The final result is:
\[ \log(a + 2b) = \log(b) + 3 \log(2) \]