How Can y^x = x^y be Solved and Visualized?

  • Thread starter cronxeh
  • Start date
In summary, the conversation discussed various mathematical functions and their solutions, particularly the equation y^x = x^y. The conversation explored different approaches to finding solutions, such as plotting and using parametric equations. It was noted that there are an infinite number of solutions and that the line y=x is not always a solution. The conversation also mentioned the existence of counterexamples, and some users proposed proofs or theories about the solutions. It was also mentioned that there is an interesting graph when plotting only in the third quadrant. Overall, the conversation touched on various aspects of mathematical problem-solving and offered different perspectives on the topic.
  • #1
cronxeh
Gold Member
1,007
11
An interesting function. If plotting you would probably omit the 0,0 point too.

How do I get the general solution for this? like y = [function of x]? Is it just y=x ?

What about the boundary conditions near origin? How does it look like there??
 
Mathematics news on Phys.org
  • #2
If you mean y being defined implicitely as a function of x, then it certainly is not "y= x". I doubt that it could be solved in terms of elementary function but you might be able to use a version the the Lambert's W function.
 
  • #3
Hello guys. I don't think it's just y=x. I've attached a rough plot of a curve I think all real solutions (in first quadrant) are on but can't prove it.

Oh yea, I think it would be an interesting exercise to figure out the complex solutions.
 

Attachments

  • sol1.JPG
    sol1.JPG
    5.6 KB · Views: 617
Last edited:
  • #4
Some particularly interesting similar function was [tex]y=x^x[/tex] When I plotted it from 0 to 1 there was what seemed like an inflection point - although I don't think its really called that but from a graph it does look pretty neat.

Could this be the plot for y^x = x^y from 0 to 1 ?
 
Last edited:
  • #5
It's not just y=x: here's a counterexample! 2^4 = 4^2
 
  • #6
Icebreaker said:
It's not just y=x: here's a counterexample! 2^4 = 4^2

there are an infinite number of counterexamples. slatydog graphed them in #3
 
  • #8
jdavel said:
there are an infinite number of counterexamples. slatydog graphed them in #3

Yes, I know. I never said it was the only counterexample, now did I?
 
  • #9
I've attached a rough plot of a curve I think all real solutions (in first quadrant) are on but can't prove it.

I think there should be a straightforward proof that there are exactly two (real) solutions for y, for each x, except for when x is... (reaches into my magic hat)... e. I just can't remember what the proof is. :biggrin:

If you happen to recall the proof that 2^y = y^2 has exactly two solutions, that should generalize to this problem.
 
  • #10
BicycleTree said:
Here's a link:
http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/x^y-x^y

Hello Bicycle. I checked out your reference. Thanks. I found it an interesting exercise to convert:

[tex]x^y=y^x[/tex]

using the substitution y=kx yielding the parametric equations:

[tex]x=k^{\frac{1}{k-1}}[/tex]

[tex]y=k^{\frac{k}{k-1}}[/tex]

However, when I plotted this, I did not obtain the line y=x. Attached is the parametric plot.
 

Attachments

  • x^y=y^x.JPG
    x^y=y^x.JPG
    3.3 KB · Views: 571
  • #11
The reason you don't get the y=x solution is that you assumed k was not 1 when you took the (k-1)-th root of the equation.
 
  • #12
my simple attempt at proving nothing...

y^x = x^y

xlogy=ylogx

(logy) = (ylogx)/x

10^(ylogx) = y

ylogxlog10 = y

logxlog10 = 1

log(x+10) = 1

10^1 = x+10

10 = x + 10

x = 0

y^0 = 0^y

y = 0

0^0 = 0^0

1 = 1

so, the only solutions I can find for y and x are = 0

(don't flame, I'm a noob and I'm probably wrong)

however...

y^0 = 0^y

0logy = ylog0 <-- that doesn't work...

0 = undefined ?
 
  • #13
y=x=0 is the only point on y=x that's not a solution.
 
  • #14
z = f(x,y) = x^y - y^x = 0

Can anyone plot this. I got some weird looking graph, maybe someone else will have a better one

Oh man this is exciting. Check out this plot!
 
Last edited:
  • #15
Given some value fixed value of x then a solution of y will be:

[tex]y = - \frac{x \text{ProductLog} \left( -\frac{\log x}{x} \right)}{\log x}[/tex]

According to mathematica. This gives you a graph like so:
 

Attachments

  • Clipboard03.jpg
    Clipboard03.jpg
    9 KB · Views: 533
  • #16
Forget the positive side its too boring! Plot all stuff in QIII !

Same interesting effect but without the positive side can be observed in [tex]x^y=-y^x[/tex]
 
Last edited:
  • #17
I think if x^y=y^x and x,y both integer then y=x

If u need the proof please tell
 
  • #18
A proof would be interesting to see, since your claim is false...
 
  • #19
https://www.physicsforums.com/attachment.php?attachmentid=3918

check that graph out
 
  • #20
I think if x^y=y^x and x,y both integer then y=x

If u need the proof please tell

[tex]2^4=4^2=16[/tex]

edit: Just noticed Icebreaker said it before me in this same thread, giving him the credit.
 
  • #21
cronxeh said:
https://www.physicsforums.com/attachment.php?attachmentid=3918

check that graph out

Holy Bernoulli, now that's interesting.
 

1. Why does y^x = x^y?

This equation is known as the "power rule" and it states that when two numbers, x and y, are raised to different powers, the result will be the same regardless of which number is the base and which is the exponent. In other words, the order of the numbers does not matter in the equation.

2. Is there a specific mathematical explanation for why this rule holds true?

Yes, there is a mathematical proof for this rule. It involves using logarithms and the properties of exponents to show that y^x = x^y. The proof can be found in most calculus textbooks.

3. How does this rule apply to real-life situations?

This rule has many real-life applications, such as in finance and economics where it is used to calculate compound interest and inflation rates. It is also used in physics and engineering to model exponential growth and decay.

4. Can this rule be extended to more than two numbers?

Yes, this rule can be extended to more than two numbers. For example, if we have three numbers, a, b, and c, and we know that a^b = b^a and b^c = c^b, then we can also say that a^c = c^a. This pattern can continue with any number of numbers.

5. Are there any exceptions to this rule?

Yes, there are exceptions to this rule. One exception is when x and y are not real numbers, but instead complex numbers. In this case, the equation does not always hold true. Additionally, this rule does not apply to negative numbers, as raising a negative number to a fractional power can result in an imaginary number.

Similar threads

I What are the applications of inverses of vector functions?
    • General Math
Replies
17
Views
2K
B Circle tangent to two lines and another circle
    • General Math
Replies
2
Views
273
MHB How Can You Rewrite an Equation to Express y as a Function of x?
    • General Math
Replies
3
Views
763
I How can Euler angles be visualized using a polar plot?
    • General Math
Replies
1
Views
1K
I A way to comprehend the geometry of a hypercube
    • General Math
Replies
2
Views
274
I Logarithmic scale - interpolation
    • General Math
Replies
2
Views
9K
B Calculating the perfect tennis shot using vectors
    • General Math
Replies
2
Views
1K
I Plotting polar equations and scale invariance
    • General Math
Replies
7
Views
1K
I Which kind of function is this?
    • General Math
Replies
3
Views
749
B Arithmetic mean of an infinite number of points?
    • General Math
Replies
20
Views
1K

Hot Threads

Recent Insights

Back
Top