Instead of Newton, bisection, analytical approximations (too many assumptions that can break during stormy weather) we do least-squares minimisation of the unimodal(!) function [$](log(x) - 5.0)^2[$] using the (most?) efficient three-point interval search on [32,243] with the least number of function evaluations. Does it do better than other algos for iv?
Here is the output (tol = 1.0e-5). Nice thing is the answer always lies in an interval.
Minimum: x, f(x) [84.75,190.25], 0.187801
Minimum: x, f(x) [111.125,163.875], 0.0467709
Minimum: x, f(x) [137.5,163.875], 0.00782745
Minimum: x, f(x) [144.094,157.281], 0.00212025
Minimum: x, f(x) [144.094,150.688], 0.000551828
Minimum: x, f(x) [147.391,150.688], 0.000139543
Minimum: x, f(x) [147.391,149.039], 3.27546e-05
Minimum: x, f(x) [147.803,148.627], 9.52942e-06
Minimum: x, f(x) [148.215,148.627], 1.93004e-06
Minimum: x, f(x) [148.318,148.524], 4.84548e-07
Minimum: x, f(x) [148.369,148.472], 1.23176e-07
Minimum: x, f(x) [148.395,148.447], 3.28335e-08
Minimum: x, f(x) [148.395,148.421], 8.72959e-09
Minimum: x, f(x) [148.408,148.421], 1.95913e-09
Minimum: x, f(x) [148.411,148.418], 5.47337e-10
Minimum: x, f(x) [148.411,148.414], 1.22007e-10
Minimum: x, f(x) [148.412,148.414], 3.37671e-11
Minimum: x, f(x) [148.413,148.414], 7.74401e-12
Minimum: x, f(x) [148.413,148.413], 2.22907e-12
Minimum: x, f(x) [148.413,148.413], 4.62369e-13
Minimum: x, f(x) [148.413,148.413], 1.17684e-13
Minimum: x, f(x) [148.413,148.413], 3.15131e-14
Minimum: x, f(x) [148.413,148.413], 8.20017e-15
Minimum: x, f(x) [148.413,148.413], 1.90432e-15
Bracket [148.413,148.413]
Bracket [2.78939e-15,1.01925e-15]
For OP, tol = 1.0e-1 gives
Minimum: x, f(x) [84.75,190.25], 0.187801
Minimum: x, f(x) [111.125,163.875], 0.0467709
Minimum: x, f(x) [137.5,163.875], 0.00782745
Minimum: x, f(x) [144.094,157.281], 0.00212025
Minimum: x, f(x) [144.094,150.688], 0.000551828
Minimum: x, f(x) [147.391,150.688], 0.000139543
Minimum: x, f(x) [147.391,149.039], 3.27546e-05
Minimum: x, f(x) [147.803,148.627], 9.52942e-06
Minimum: x, f(x) [148.215,148.627], 1.93004e-06
Minimum: x, f(x) [148.318,148.524], 4.84548e-07
Minimum: x, f(x) [148.369,148.472], 1.23176e-07
Bracket [148.369,148.472]
Bracket [8.70207e-08,1.59332e-07]