Let' s have a look at Newton' s method for a few functions and starting points :

(* This is Newton's method: *)
Newton[f_,x0_]:=x0-f[x0]/f'[x0]

Clear[f]
f[x_]:=Log[x]−1.23
x0=0.5
x1=Newton[f,x0]
x2=Newton[f,x1]
x3=Newton[f,x2]
x4=Newton[f,x3]
Solve[f[x]==0,x]
p1=Plot[f[x],{x,1,4}];
p2=ListPlot[{{x0,f[x0]},{x1,f[x1]},{x2,f[x2]},{x3,f[x3]},{x4,f[x4]}},PlotMarkers->Automatic];
Show[p1,p2]

Clear[f]
f[x_]:=Cos[x]−.23
x0=1.2
x1=Newton[f,x0]
x2=Newton[f,x1]
x3=Newton[f,x2]
x4=Newton[f,x3]
Solve[f[x]==0,x]
p1=Plot[f[x],{x,1,1.5}];
p2=ListPlot[{{x0,f[x0]},{x1,f[x1]},{x2,f[x2]},{x3,f[x3]},{x4,f[x4]}},PlotMarkers->Automatic];
Show[p1,p2]

Clear[f]
f[x_]:=x(x^2−1)
x0=.4
x1=Newton[f,x0]
x2=Newton[f,x1]
x3=Newton[f,x2]
x4=Newton[f,x3]
Solve[f[x]==0,x]
p1=Plot[f[x],{x,-1,1}];
p2=ListPlot[{{x0,f[x0]},{x1,f[x1]},{x2,f[x2]},{x3,f[x3]},{x4,f[x4]}},PlotMarkers->Automatic];
Show[p1,p2]