If a, b, c are positive real numbers, then the minimum value of
`a^(logb-logc)+b^(logc-loga)+c^(loga-logb)`is

Options

A. 3
B. 1
C. 9
D. 16

Answer

To find the minimum value of the expression \( a^{(\log b - \log c)} + b^{(\log c - \log a)} + c^{(\log a - \log b)} \), we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality. Here’s a step-by-step solution:

### Step 1: Apply AM-GM Inequality
Using the AM-GM inequality, we know that for any positive real numbers \( x_1, x_2, \ldots, x_n \):
\[ \frac{x_1 + x_2 + \ldots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \ldots x_n} \]
In our case, we can apply this to the three terms in our expression.

### Step 2: Set Up the Expression
Let:
\[ x_1 = a^{(\log b - \log c)}, \quad x_2 = b^{(\log c - \log a)}, \quad x_3 = c^{(\log a - \log b)} \]
Then, by AM-GM:
\[ \frac{x_1 + x_2 + x_3}{3} \geq \sqrt[3]{x_1 x_2 x_3} \]

### Step 3: Calculate the Product
Now, we need to calculate the product \( x_1 x_2 x_3 \):
\[ x_1 x_2 x_3 = a^{(\log b - \log c)} \cdot b^{(\log c - \log a)} \cdot c^{(\log a - \log b)} \]

### Step 4: Simplify the Product
Using properties of logarithms:
\[ x_1 x_2 x_3 = a^{\log b - \log c} \cdot b^{\log c - \log a} \cdot c^{\log a - \log b} \]
This can be rewritten as:
\[ = \frac{a^{\log b}}{a^{\log c}} \cdot \frac{b^{\log c}}{b^{\log a}} \cdot \frac{c^{\log a}}{c^{\log b}} \]
This simplifies to:
\[ = \frac{a^{\log b} \cdot b^{\log c} \cdot c^{\log a}}{a^{\log c} \cdot b^{\log a} \cdot c^{\log b}} \]

### Step 5: Take Logarithm
Taking logarithm on both sides:
\[ \log(x_1 x_2 x_3) = \log(a^{\log b}) + \log(b^{\log c}) + \log(c^{\log a}) - \left(\log(a^{\log c}) + \log(b^{\log a}) + \log(c^{\log b})\right) \]
This will lead to cancellation of terms, ultimately resulting in:
\[ \log(x_1 x_2 x_3) = 0 \]
Thus, \( x_1 x_2 x_3 = 1 \).

### Step 6: Apply AM-GM Result
Now substituting back into the AM-GM inequality:
\[ \frac{x_1 + x_2 + x_3}{3} \geq \sqrt[3]{1} = 1 \]
This implies:
\[ x_1 + x_2 + x_3 \geq 3 \]

### Conclusion
The minimum value of \( a^{(\log b - \log c)} + b^{(\log c - \log a)} + c^{(\log a - \log b)} \) is therefore:
\[ \boxed{3} \]