Question

Consider two distinct positive numbers \( m, n \) with \( m > n \). Let \[ x = n^{\log_n m}, \quad y = m^{\log_m n}. \] The relation between \( x \) and \( y \) is - 
 

• A. x> y
• B. x = y
• C. x = log$_{10}$ y
• D. x< y

Hint:

Always look to simplify expressions involving logarithms and exponents by applying fundamental identities. The identity $a^{\log_a b} = b$ is one of the most important and frequently tested properties.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given two expressions, x and y, defined using logarithms and exponents. We need to simplify them and determine the relationship between them, given that m> n.
Step 2: Key Formula or Approach:
The fundamental property of logarithms that applies here is:
\[ a^{\log_a b} = b \] This identity states that exponentiation and logarithm are inverse operations when the base of the exponent and the base of the logarithm are the same.
Step 3: Detailed Explanation:
Let's simplify the expression for x:
\[ x = n^{\log_n m} \] Using the identity $a^{\log_a b} = b$, with $a=n$ and $b=m$, we get:
\[ x = m \] Now, let's simplify the expression for y:
\[ y = m^{\log_m n} \] Using the same identity, with $a=m$ and $b=n$, we get:
\[ y = n \] So we have found that $x = m$ and $y = n$.
The problem states that m and n are distinct positive numbers with the condition $m>n$.
Substituting our simplified expressions for x and y into this inequality, we get:
\[ x>y \] Step 4: Final Answer:
The relation between x and y is x> y.