It seems the real solution of the following equation $$t=u^2+u\log(u) $$ has no closed form in view of the output of



```
Reduce[t == u^2 + u*Log[u], u, Reals]

```

The question arises about the asymptotics of that solution as $t$ tends to infinity.



Asymptotics of solution of transcendental equation
Your equation has the form
$$u=\sqrt{t-u\log u}.$$
First obvious approximation is $u\_0(t)=\sqrt{t}$, second approximation is
$$u\_1(t)=\sqrt{t-u\_0(t)\log u\_0(t)}=\sqrt t-\frac{\log t}4+O\left(\frac{\log^2t}{\sqrt t}\right).$$ Next one is $u\_2(t)=\sqrt{t-u\_1(t)\log u\_1(t)}.$


