If x^y = y^x and y= 2x find x

Answer 1

We start this by taking natural log on both sides of second equation:

ln(y) = ln(2x)

Using: ln(MN) = log(M) +ln(N), we have: ln(y) = ln(2) + ln(x)............ (1)

Doing the same for equation 1.

ln(y^x) = ln(x^y)

Using: ln(Mⁿ) = n*ln(M), we have: x*ln(y) = y*ln(x)................(2)

Substituting all y terms in terms of x:

x*(ln(2) + ln(x)) = 2x*(ln(x))

ln(2) + ln(x) = 2* ln(x)

thus, ln(x) = ln(2) gives: x = 2 and y = 4.