Question

Define one relation that depicts the following scenario:

Find set of numbers, x and y, such that x raised to y is n, and y raised to x is the reciprocal of n.

Answer 1

I get that
\(x^{y}=n\)
and
\(y^{x}=\frac{1}{n}\)
are the two relations but I need to make it one that shows the possible values of x and y??

Answer 2

It looks very difficult to do but once you realize the trick then it's easy.

Use substitution.
You know that \(x^{y}=n\) and \(y^{x}=\frac{1}{n}\), but you also know that the n is the same. Use the mathematical property of substitution.
\(x^{y}=\frac{1}{y^{x}}\)
Now it's just simplifying.
\(x^{y}=y^{-x}\) \(\ln\left(x^{y}\right)=\ln\left(y^{-x}\right)\) \(y\ln x=-x\ln y\) \(\frac{y}{\ln y}=-\frac{x}{\ln x}\) and poof, there's your relation!!

Answer 3

Probably they'll except your answer as this:
\(y=-\frac{x\ln y}{\ln x}\)

Answer 4

Thanks so much man!!!!! I actually get it! Can I post more in the new questions?