If `x gt 0`, `y gt 0`, `z gt 0`, the least value of
`x^(log_(e)y-log_(e)z)+y^(log_(e)z-log_(e)x)+Z^(log_(e)x-log_(e)y)` is

Options

A. `3`
B. `1`
C. `5`
D. `6`

Explanation

To find the least value of the expression

\[ x^{\log_e y - \log_e z} + y^{\log_e z - \log_e x} + z^{\log_e x - \log_e y} \]

given that \(x > 0\), \(y > 0\), and \(z > 0\), we can follow these steps:

### Step 1: Change of Variables
Let us define:
- \(a = \log_e x\)
- \(b = \log_e y\)
- \(c = \log_e z\)

This transforms our expression into:

\[ e^{a(b - c)} + e^{b(c - a)} + e^{c(a - b)} \]

### Step 2: Applying the AM-GM Inequality
We can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality to the three terms:

\[ \frac{e^{a(b - c)} + e^{b(c - a)} + e^{c(a - b)}}{3} \geq \sqrt[3]{e^{a(b - c)} \cdot e^{b(c - a)} \cdot e^{c(a - b)}} \]

### Step 3: Simplifying the Right Side
The product on the right side can be simplified:

\[ e^{a(b - c) + b(c - a) + c(a - b)} \]

### Step 4: Expanding the Exponent
Now, let’s expand the exponent:

\[ a(b - c) + b(c - a) + c(a - b) = ab - ac + bc - ba + ca - cb = 0 \]

### Step 5: Conclusion from AM-GM
Thus, we have:

\[ \sqrt[3]{e^0} = \sqrt[3]{1} = 1 \]

So, we can conclude:

\[ e^{a(b - c)} + e^{b(c - a)} + e^{c(a - b)} \geq 3 \]

### Final Result
The least value of the expression is therefore:

\[ \boxed{3} \]