Just wondering about x^y = y^x

• Sink41
In summary, the conversation discusses the equation y^x = x^y and its graphical representation. The speaker mentions two questions about this equation and the possibility of writing it as y = f(t) and x = g(t). They also mention the possibility of finding the equation of a line that resembles y = e^(e/x), but express their difficulty in solving these questions without knowledge of higher level mathematics. They mention using a graphing program, specifically a trial version of Derive 6, to plot the equation and discuss its accuracy.

Sink41

This wasnt set for homework just something i thought up so i hope this is the right place to post?

Anyway when i put x^y = y^x (or also lnx/lny = x/y) in a graphing program i realized that there were two lines, y = x and another line that looked a bit like y = e^(e/x) but wasnt.

So this kind of interested me... and i have two questions:

can y^x = x^y be written as y = f(t) and x = g(t) and if so what are f(t) and g(t)?

Is it possible to find the equation of just the line that looks like y = e^(e/x) ?

I have thought about it a bit but have no idea how to go about solving either question. Any degree level maths will be beyond me (doing A level atm) so hopefully there is something simple that i have overlooked?

How did you put xy= yx into a graphing program? Most graphing programs require that the function be put in the form y= f(x) which is the whole problem here.

Solving yx= xy, or, equivalently, y= xy/x for y looks to me like it would involve the "Lambert W function", defined as the inverse to the function f(x)= xex.

I used a trial version of derive 6 and plugged x^y=y^x straight in. It only plotted for positive values and went a bit crazy at high values of x and y. I think it was fairly accurate though. I remember putting one of the values for the non x=y line into my calculator and it working.

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

In this equation, x and y are both variables that represent any real numbers. This means that the values of x and y can be any positive or negative number, including fractions and decimals.

2. What does the caret symbol (^) mean in x^y = y^x?

The caret symbol in this equation is the exponentiation operator, which means that the number or variable before it is raised to the power of the number or variable after it. In other words, x^y is equivalent to x raised to the power of y.

3. Can x and y be equal in x^y = y^x?

Yes, x and y can be equal in this equation. When x and y are equal, the equation becomes x^x = x^x, which is always true for any value of x. This is because any number raised to the power of itself is always equal to itself.

4. What is the significance of x^y = y^x in mathematics?

This equation is known as an exponential equation and holds significant importance in mathematics. It is often used to model real-world scenarios, such as population growth and radioactive decay. It also has many applications in calculus and other branches of mathematics.

5. Are there any solutions to x^y = y^x where x and y are not equal?

Yes, there are infinite solutions to this equation where x and y are not equal. These solutions can be found by using logarithms and solving for either x or y. For example, if x = 2, then y = 4 is a solution, and if x = 3, then y = 9 is another solution. However, these are just two of the many possible solutions.