To find cube roots of any number with a simple calculator, the following method was given to us by our teacher, which is accurate to atleast one-tenths.
1)Take the number $X$, whose cube root needs to be found out, and take its square root 13 times (or 10 times) i.e. $\sqrt{\sqrt{\sqrt{\sqrt{....X}}}}$
2)next, subtract $1$, divide by $3$ (for cube root, and any number $n$ for $n$th root), add $1$.
3) Then square the resultant number (say $c$) 13times (or 10 times if you had taken out root 10 times) i.e. $c^{2^{2^{....2}}}=c^{2^{13}}$. This yields the answer.
I am not sure whether taking the square root and the squares is limited to 10/13 times, but what I know is this method does yield answers accurate to atleast one-tenths.
For finding the log, the method is similar:-
1)Take 13 times square root of the number, subtract 1, and multiply by $3558$. This yield s the answer.
Why do these methods work? What is the underlying principle behind this?