What is the best method to use to find the root of a log shaped function?

Question

I am trying to find the best method to get the root of a log shaped function. I have a function which is shaped very much like $y=ln(x)$. That is the part to the left of the zero is very steep and has large positive slope, while the part to the right has a slope approaching zero. Using Newton-Raphson on the right of the root results in estimate too far to left for which there is no solution. Using NR on the left would work just fine, but this region is difficult to find as the domain of the function here is very small.