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The two equations alow a quick elimination of $x>0$. We can than numerically solve the remaining equation in the variable $y$. The following code may help to get the relevant information, adapted to my taste:

sage: f(y) = log(y) +1/y -1 - log(2)
sage: f
y |--> 1/y - log(2) + log(y) - 1
sage: plot( f, 0.2, 0.5 )
Launched png viewer for Graphics object consisting of 1 graphics primitive
sage: find_root( f, 0.2, 0.4 )
0.3733646177016173
sage: gp( "solve( y=0.2, 0.4, %s )" %f(y) )
0.37336461770167408424844843667927059501


(The value for $x$ is easily found from the second equation.)

The two equations alow allow a quick elimination of $x>0$. We can than numerically solve the remaining one equation in the remaining single variable $y$. The following code may help to get the relevant information, adapted to my taste:

sage: f(y) = log(y) +1/y -1 - log(2)
sage: f
y |--> 1/y - log(2) + log(y) - 1
sage: plot( f, 0.2, 0.5 )
Launched png viewer for Graphics object consisting of 1 graphics primitive
sage: find_root( f, 0.2, 0.4 )
0.3733646177016173
sage: gp( "solve( y=0.2, 0.4, %s )" %f(y) )
0.37336461770167408424844843667927059501


(The value for $x$ is easily found from the second equation.) equation.)