If `log ((a + b)/(2)) = (1)/(2) (log a + log b)`, then a is equal to

Options

A. 0
B. `-b`
C. b
D. None of these

Answer

To solve the equation \( \log \left( \frac{a + b}{2} \right) = \frac{1}{2} (\log a + \log b) \), we will follow these steps:

### Step 1: Simplify the Right-Hand Side
The right-hand side can be simplified using the property of logarithms that states \( \log x + \log y = \log(xy) \).

\[ \frac{1}{2} (\log a + \log b) = \frac{1}{2} \log(ab) = \log(ab)^{1/2} = \log(\sqrt{ab}) \]

### Step 2: Set the Two Sides Equal
Now we can rewrite the equation as:

\[ \log \left( \frac{a + b}{2} \right) = \log(\sqrt{ab}) \]

### Step 3: Remove the Logarithm
Since the logarithm function is one-to-one, we can eliminate the logarithm by exponentiating both sides:

\[ \frac{a + b}{2} = \sqrt{ab} \]

### Step 4: Multiply Both Sides by 2
To eliminate the fraction, multiply both sides by 2:

\[ a + b = 2\sqrt{ab} \]

### Step 5: Rearrange the Equation
Now, we can rearrange the equation:

\[ a + b - 2\sqrt{ab} = 0 \]

### Step 6: Recognize a Perfect Square
This expression can be recognized as a perfect square:

\[ (a - \sqrt{ab})^2 = 0 \]

### Step 7: Solve for \( a \)
Taking the square root of both sides gives us:

\[ a - \sqrt{ab} = 0 \]

Thus,

\[ a = \sqrt{ab} \]

### Step 8: Square Both Sides
Squaring both sides yields:

\[ a^2 = ab \]

### Step 9: Rearrange to Find \( a \)
Rearranging gives:

\[ a^2 - ab = 0 \]

Factoring out \( a \):

\[ a(a - b) = 0 \]

### Step 10: Solve the Factored Equation
This gives us two solutions:

1. \( a = 0 \)
2. \( a = b \)

### Conclusion
Thus, the value of \( a \) can be either \( 0 \) or \( b \).

### Final Answer
The answer is \( a = b \).

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