System of equation

[picio]

Hello,

I have a system to solve and I am really having difficulties to do it
ln(e^x+e^y)=5
e^(ln(x)/ln(y))=1/e

Thanks for your help
Picio

[HallsofIvy]

Hello,

I have a system to solve and I am really having difficulties to do it
ln(e^x+e^y)=5
e^(ln(x)/ln(y))=1/e

Thanks for your help
Picio

1/e= e^{-1} so e^(ln(x)/ln(y))= e^{-1} and, since the exponential function is "one to one", ln(x)/ln(y)= -1. So ln(x)= -ln(y)= ln(1/y) and, since the logarithm function is also "one to one", x= 1/y.

From ln(e^x+ e^y)= 5, e^x+ e^y= e^5. Since x= 1/y, y= 1/x and e^x+ e^(1/x)= e^5. At this point, I'm stuck! There is no simple relationship between "e^x" and "e^(1/x)".