X to the xth power and other indices

In summary, the conversation discusses two functions xy = yx and y = xx and their solutions. The first function has a solution of y = x and the second function has a minimum value of x at approximately 0.45. It is also mentioned that there may not be an elementary closed form solution for these functions.
  • #1
jcsd
Science Advisor
Gold Member
2,115
13
Recently I've considering the two functions xy= yx and y = xx.

1) For xy= yx can y be found in terms of x? I'm sure I've seen such a solution before.

2) In y = xx when y (and therefore x)is postive what value of y gives the minium value for x? I know it's rougly 0.45 and that dy/dx = 0 at this point, but it's a long time since I've done more advanced differentiation and I don't even know if you can differentiate xx
 
Mathematics news on Phys.org
  • #2
2)

y(x) = xx
ln y(x) = x ln x

(1 / y(x)) * y'(x) = 1 * ln x + x (1 / x)
y'(x) = y(x) (1 + ln x)
y'(x) = xx(1 + ln x)

Since xx > 0, y'(x) can be 0 iff
1 + ln x = 0
ln x = -1
x = 1/e

So the minimum of xx occurs at x = 1/e = 0.368


1)

y = x

Ok, ok, that's not the only solution. I highly doubt there is an elementary solution for this, but I can get you started:

for x, y > 0:

xy = yx
y ln x = x ln y
y / ln y = x / ln x
ln y / y = ln x / x

(note each step is reversible)

so consider f(t) = ln t / t
then f'(t) = (1 - ln t) / t2

So we see that f(t) is strictly increasing for t < e and strictly decreasing for t > e. This means that for any z, f(t)=z has at most two solutions. More specifically one can fairly easily show that:

the equation f(t) = z has:
exactly 1 solution if z <= 0
exactly 2 solutions if 0 < z < 1/e
exactly 1 solution if z = 1/e
exactly 0 solutions if z > 1/e


We've seen that if xy = yx iff f(x)=f(y).

y = x is clearly a solution... meaning that it is the only solution (for a given x) iff f(x) <= 0 or f(x) = 1/e... that is if x <= 1 or x = e.

Also, we can see that if there are two solutions of xy = yx for a given x, then either
1 < x < e < y
or
1 < y < e < x

For example, the only nontrivial solution I know off the top of my head is 24 = 42... clearly 1 < 2 < e < 4


But as I mentioned, I strongly suspect no elementary closed form solution is possible.
 
  • #3
I believe it is

y(x) = (x productlog(-ln(x)/x))/ln(x);
where prductlog(x) gives the principal solution for w in x=w e^w.
Not all solutions are found.
 
  • #4
well, i am not quite sure about it so don't take it for granted...
 

1. What is the meaning of "X to the xth power"?

X to the xth power is a mathematical expression that indicates the number X being multiplied by itself x number of times. For example, 2 to the 3rd power (2^3) means 2 multiplied by itself 3 times, which is equal to 8.

2. How do you calculate X to the xth power?

To calculate X to the xth power, you can use the exponent rule which states that X^a * X^b = X^(a+b). This means that you can add the exponents when multiplying powers with the same base. For example, 2^3 * 2^2 = 2^(3+2) = 2^5 = 32.

3. What is the difference between X to the xth power and X to the yth power?

The difference between X to the xth power and X to the yth power is the value of the exponent. In X to the xth power, the exponent is a specific number or value, while in X to the yth power, the exponent can vary and can be any real number or variable.

4. How do you simplify expressions with indices?

To simplify expressions with indices, you can use the properties of exponents. Some common properties include the power of a product rule, power of a quotient rule, and power of a power rule. These properties allow you to simplify expressions by combining like terms or breaking down larger powers into smaller ones.

5. How is the concept of indices used in real life?

The concept of indices is used in various fields such as finance, science, and engineering. In finance, indices are used to track and measure the performance of stocks and investments. In science, indices are used to represent quantities such as time, distance, and energy. In engineering, indices are used to simplify complex calculations and equations.

Similar threads

  • General Math
Replies
3
Views
769
  • Calculus and Beyond Homework Help
Replies
1
Views
978
  • General Math
Replies
10
Views
2K
  • General Math
2
Replies
41
Views
4K
  • Atomic and Condensed Matter
Replies
8
Views
3K
  • General Math
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Science and Math Textbooks
Replies
5
Views
2K
Replies
1
Views
796
Back
Top