In[11]:= Integrate[Log[x]/Log[17/20], {x, 0, x2}]
Solve[% == 1/2 (% /. x2 -> 1), x2]
% // N
Out[11]= (x2 - x2 Log[x2])/Log[20/17]
During the evaluation of In[11]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
Out[12]= {{x2 -> -(1/(2 ProductLog[-(1/(2 E))]))}, {x2 -> -(1/(
2 ProductLog[-1, -(1/(2 E))]))}}
Out[13]= {{x2 -> 2.15554}, {x2 -> 0.186682}}
Basically, the curve is a scaled version of -Log[x].
In[22]:= Integrate[-Log[x], {x, 0, x2}]
Solve[% == 1/2 (% /. x2 -> 1), x2]
% // N
Out[22]= x2 - x2 Log[x2]
During the evaluation of In[22]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
Out[23]= {{x2 -> -(1/(2 ProductLog[-(1/(2 E))]))}, {x2 -> -(1/(
2 ProductLog[-1, -(1/(2 E))]))}}
Out[24]= {{x2 -> 2.15554}, {x2 -> 0.186682}}