Exploring Solutions of x^y=y^x: A Deeper Look at the Equation

  • Thread starter Ali Asadullah
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In summary, the solutions to x^y=y^x are (2,4) and (4,2), but there are also other solutions such as (3,3) and (4,4) with certain restrictions. Additionally, there are complex solutions as well. The thread provided offers more information on this topic.
  • #1
Ali Asadullah
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We know that the solutions of x^y=y^x are (2,4) and (4,2), but please tell me how to solve it. I have tried to take log wrt x on both side.
 
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  • #3
Ali Asadullah said:
We know that the solutions of x^y=y^x are (2,4) and (4,2)

This is only the case for when restricted to x=2 or y=4. Another example would be (3,3) and (4.4) - of course this could be generalised to all y=x with certain restrictions.

But this isn't the only set of solutions, Taking (3,y) we can have y=3 and also [itex]y\approx 2.478[/itex].
And if you take a look at the thread arildno posted, you'll see there are complex solutions as well.
 

FAQ: Exploring Solutions of x^y=y^x: A Deeper Look at the Equation

1. What is the equation x^y = y^x?

The equation x^y = y^x is a form of exponential equations where both the base and the exponent are variables. This equation is also known as the power function or the exponential equality and it has infinite solutions.

2. How do you solve the equation x^y = y^x?

The equation x^y = y^x can be solved by taking the logarithm of both sides. This will result in a linear equation that can be solved using algebraic methods.

3. What are the solutions for the equation x^y = y^x?

The equation x^y = y^x has infinite solutions. However, when given specific points, there are only two solutions that satisfy the equation. In the case of (2,4) and (4,2), the solutions are x = 2 and x = 4.

4. How do you verify the solutions for the equation x^y = y^x?

To verify the solutions for the equation x^y = y^x, simply plug in the values for x and y into the equation. If the equation holds true, then the solution is valid.

5. What is the significance of the solutions for the equation x^y = y^x at (2,4) and (4,2)?

The solutions for the equation x^y = y^x at (2,4) and (4,2) represent the point where the two exponential curves intersect. These points are known as the "golden points" and they have a special property where the slopes of the curves at these points are equal.

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