If p, q, r are + ive, then the minimum value of `p^(log q - log r) + q^(log r - log p) + r^(log p - log q)` is

Options

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

Answer

To find the minimum value of the expression \( p^{(\log q - \log r)} + q^{(\log r - \log p)} + r^{(\log p - \log q)} \), we can follow these steps:

### Step 1: Rewrite the expression using properties of logarithms
We can use the property that \( a^{\log_b c} = c^{\log_b a} \). Thus, we can rewrite each term in the expression:
- \( p^{(\log q - \log r)} = \frac{p^{\log q}}{p^{\log r}} = \frac{q^{\log p}}{r^{\log p}} \)
- \( q^{(\log r - \log p)} = \frac{q^{\log r}}{q^{\log p}} = \frac{r^{\log q}}{p^{\log q}} \)
- \( r^{(\log p - \log q)} = \frac{r^{\log p}}{r^{\log q}} = \frac{p^{\log r}}{q^{\log r}} \)

### Step 2: Apply the AM-GM inequality
Now, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{p^{(\log q - \log r)} + q^{(\log r - \log p)} + r^{(\log p - \log q)}}{3} \geq \sqrt[3]{p^{(\log q - \log r)} \cdot q^{(\log r - \log p)} \cdot r^{(\log p - \log q)}} \]

### Step 3: Simplify the product
The product \( p^{(\log q - \log r)} \cdot q^{(\log r - \log p)} \cdot r^{(\log p - \log q)} \) can be simplified:
\[ = p^{\log q} \cdot p^{-\log r} \cdot q^{\log r} \cdot q^{-\log p} \cdot r^{\log p} \cdot r^{-\log q} \]
This simplifies to:
\[ = \frac{p^{\log q} \cdot q^{\log r} \cdot r^{\log p}}{p^{\log r} \cdot q^{\log p} \cdot r^{\log q}} \]

### Step 4: Use the equality condition of AM-GM
The AM-GM inequality states that the equality holds when all terms are equal. Therefore, we set:
\[ p^{(\log q - \log r)} = q^{(\log r - \log p)} = r^{(\log p - \log q)} \]
This leads us to conclude that \( p = q = r \).

### Step 5: Substitute equal values
Let \( p = q = r = k \) (where \( k \) is a positive constant). The expression becomes:
\[ 3 \cdot k^{(\log k - \log k)} = 3 \cdot k^0 = 3 \]

### Conclusion
Thus, the minimum value of the expression \( p^{(\log q - \log r)} + q^{(\log r - \log p)} + r^{(\log p - \log q)} \) is \( 3 \).