Dtyjlui

How would I find the roots of this function?

begin{equation}
x^2 - frac{1}{ln(x) - 1} + frac{1}{x^2(ln(x)-1)} = 0
end{equation}

I don't really know how i would tackle this problem because we're working with natural logarithms. Does anyone have an idea?

If you plot the fuction, you see solutions around $x=1$ and $x=3$ from which you can start Newton method.

You can have approximations using series expansions.

Around $x=0$,
$$x^2 - frac{1}{ln(x) - 1} + frac{1}{x^2(ln(x)-1)} = 1+4 (x-1)+Oleft((x-1)^3right)$$ and then the solution is close to
$$x=frac{3}4$$

Around $x=3$, the expansion would be
$$left(9-frac{8}{9 (log (3)-1)}right)+frac{2 (x-3) left(86+81 log ^2(3)-163
log (3)right)}{27 (log (3)-1)^2}+Oleft((x-3)^2right)$$ and the the solution is close to
$$x=frac{3 left(83+81 log ^2(3)-156 log (3)right)}{2 left(86+81 log ^2(3)-163
log (3)right)}approx 3.00039$$

Edit

With regard to the first root, you could have better and better rational approximations building the $[1,n]$ Padé approximant. This would give
$$left(
begin{array}{ccc}
n & x_{(n)} & x_{(n)} approx \
1 & frac{3}{4} & 0.750000 \
2 & frac{25}{33} & 0.757576 \
3 & frac{1247}{1643} & 0.758977 \
4 & frac{5179}{6822} & 0.759162 \
5 & frac{11947}{15737} & 0.759166 \
6 & frac{1339361}{1764260} & 0.759163
end{array}
right)$$

Hint: By a numeric method we find $$xapprox 0.75916167335311382164$$ or $$xapprox 3.00039175573599420937$$

I usually do a quick plot like Plot[x^2-1/(Log[x]-1)+1/(x^2(Log[x]-1)),{x,-10,+10}] which shows the above roots quickly and that there are no others.