What is the answer of fallowing equation?

Question

2^x=3^y
x^y = y^x

x=?
y=?

Answer 1

Best Answer - Chosen by Asker

2^x=3^y >>> x ln(2) = yln(3)>> x= y ln(3)/ln(2) Eq1

x^y = y^x >>> y ln(x) = xln(y) ......Eq2

from Eq1 in Eq2 :

y ln(x) = xln(y)
y ln[yln(3)/ln(2)]= [yln(3)/ln(2)] * ln(y)
y ln[y ln(3)/ln(2)] = y ln(3)/ln(2) *ln(y) ...Eq3
as y is not zero (( and can not be zero ))
we divide Eq3 on y:
ln[yln(3)/ln(2)]=ln(3)/ln(2)*ln(y)
ln(y)+ln(ln(3)/ln(2))= ln(3)* ln(y) /ln(2)
ln(y)[ln(3)/ln(2)-1] = ln(ln(3)/ln(2))
ln(y)=ln(ln(3)/ln(2))/[ln(3)/ln(2)-1]
ln(y) = 0.79 >>> y= 2.20

from Eq1 : x= y ln(3)/ln(2) = 3.48

To be sure x^y=y^x = 15.52
2^x= 3^y= 11.18


Nice question , Thank you , Very interesting .

Notice : that we divded on y in Eq3 and y can not be zero , and if y=0 >>> 2^x=3^0 =1 >>> x=0 and in this case x^y= y^x = Unknown answer as 0^0 is unknown .


Best regards

Asker's Rating:

Asker's Comment:

thanks ,
very good!

Answer 2

x=y