The equation given is:

$x^y = y^x$

Step 1: Consider Simple Cases

• Case 1: If $x = y$, then obviously both sides are equal.
• Case 2: If $x = y = 2$, then $2^2 = 2^2$, which is true.
• Case 3: If $x = 4, y = 2$, then:
  $4^2 = 16, \quad 2^4 = 16$ This holds true.

Step 2: General Approach

Taking the natural logarithm on both sides:
$\ln(x^y) = \ln(y^x)$

Using the logarithm rule $\ln(a^b) = b\ln a$, we get:
$y \ln x = x \ln y$

Rearrange:
$\frac{\ln x}{x} = \frac{\ln y}{y}$

This equation suggests that $x$ and $y$ must be values where the function $f(t) = \frac{\ln t}{t}$ is equal for both $x$ and $y$.

Step 3: Finding Solutions

1. Trivial solution: $x = y$
2. Non-trivial integer solution: $(x, y) = (4,2)$ and $(2,4)$
3. General solution: Solutions exist where $\frac{\ln x}{x} = \frac{\ln y}{y}$, which can be solved graphically or numerically.

Would you like further elaboration or graphical solutions?

Related Questions:

1. How can we find more integer solutions to $x^y = y^x$?
2. Can there be non-integer solutions to this equation?
3. How does the function $f(t) = \frac{\ln t}{t}$ help in finding solutions?
4. What happens when $x$ or $y$ is a fraction?
5. Can we extend this equation to complex numbers?

Tip:

Graphing $f(t) = \frac{\ln t}{t}$ can help visualize solutions where two different values of $t$ yield the same function value.