Prove that if xy yx then lnxx lnyySolutionIt is given that

Prove that if x^y = y^x, then ln(x)/x = ln(y)/y.

Solution

It is given that x^y = y^x

Take the logarithm to the base e, or ln, of both the sides

=> ln [ x^y] = ln [ y^x]

Use the property of log that log a^b = b*log a

=> y*ln x = x*ln y

divide both the sides by xy

=> [y*ln x]/xy = [x*ln y]/xy

=> (ln x)/x = (ln y)/y

If x^y = y^x, then this proves that (ln x)/x = (ln y)/y