Solving System of equations: ln(e^x+e^y)=5 and e^(ln(x)/ln(y))=1/e

[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

 

[Deleted member 4993]

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

Start with:

1/e = e^(-1)

and continue....

 

[picio]

Start with:

1/e = e^(-1)

and continue....

I did this
I now have e^x+e^y=e^5, ln(x)/ln(y)=-1

Then I simplified
ln(x)/ln(y)=-1
ln(x)=-ln(y)
ln(x)=ln(1/y)
x=1/y

And I am stuck here

 

[picio]

That's what I did
I have e^x+e^y=e^5 and ln(x)/ln(y)=-1
Then I simplified and found x=1/y and I'm blocked here

 

[JeffM]

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

SK gave you one hint. I shall give you another.

\(\displaystyle ln(e^x + e^y) = 5.\)

\(\displaystyle BUT\ 5 = 5 * 1 = 5 * ln(e) = WHAT?\)

 

[Steven G]

That's what I did
I have e^x+e^y=e^5 and ln(x)/ln(y)=-1
Then I simplified and found x=1/y and I'm blocked here

Replace 1/y for x in e^x+e^y=e^5