TVM solve for interest rate, revisited

[Albert Chan]

Found a nice rate formula! https://brownmath.com/bsci/loan.htm#LoanInterest
Any clue how q-to-rate method is derived? Why log2?

fv = 0
i0 = pmt/-pv
q = log2(1+1/n)                    → i ≈ ((1+i0)^(1/q)-1)^q-1

Or, let Q = 1/q ≈ n*log(2)      → (1+i)^Q ≈ (1+i0)^Q - 1

Example from previous post

lua> n, pv, pmt, fv = 525600, 0, -1, 526038.2426000326
lua> 1/(600*n)                  -- true rate
3.1709791983764586e-09

lua> n, pv, fv = -n, -fv, -pv -- time reversal to match formula
lua> i0 = pmt/-pv
lua> q = log2(1+1/n)
lua> EFF(APR(i0,q)-1, q)
3.1710141058643196e-09

lua> i0 - 1/(i0*n^2)           -- compare to I ≈ 1/P - P/N^2
3.1714139412113494e-09