Studying the equation $\ln(x)- 1+ x^2=0$

Question

Can the equation $\ln(x)- 1 + x^2=0$ be solved without the Lambert $W$ function ? (I didn't study Lambert W yet).

By inspection I can see that $x=1$ is a solution. Are there other solutions? And is there a systematic way to find them?

Answer 1

The function $f(x):=\ln x-1+x^2$ on $x>0$ satisfies $\lim_{x\to0^+}f(x)=-\infty,\,\lim_{x\to\infty}f(x)=\infty$ and $f^\prime=\frac1x+2x>0$, so the root on $\Bbb R^+$ is unique. By inspection, it's $x=1$.

Edit, since the function whose roots are sought has been changed to $g(x):=\ln x+1-x^2$: it's unclear the root $\ne1$ can be obtained analytically without Lambert $W$, but you could numerically approximate it, e.g. with Newton-Raphson viz.$$x_0=\frac12,\,x_{n+1}=x_n-\frac{\ln x_n+1-x_n^2}{1/x_n-2x_n}.$$[Reverted]