Formulating goal seeking tool - a challenging problem

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You might find this tutorial of mine interesting: https://www.excelforum.com/tips-and-...ind-roots.html In it, I show hot to set up the spreadsheet to use NR and secant algorithms (and circular references) to solve these kinds of problems. Adapting my polynomial examples to your function should be straightforward.

Since I have only limited literacy in the mathematical world

I find this unfortunate, because I find that I must first understand the math behind a problem before I can reliably program the solution. Have you had any success understanding how Newton's method works?

Would the process easier or complicated than manual goal-seeking?

It appears that you already know how to manually use Goal-seek to solve this problem, so I would say that it will be easier -- though quite tedious -- to do it manually rather than try to program a more automated solution. I find that this usually feeds into some kind of "cost-benefit" analysis. Is the effort/cost required for programming a less tedious solution offset by the benefit of less tedium. Only you can decide for yourself.

is there a possible excel equation that can be developed?

You noted in your OP, and I would concur, that this function does not have an easy inverse function. Excel formulas are not magic -- if the formula cannot be inverted, there will not be a simple Excel formula for the inverse, either. As the final example (post #9) in my tutorial shows, one could perform the root finding algorithm in a VBA UDF, which will simplify things in the spreadsheet. It's the same algorithm, just moved into a different programming language.

Do I need to develop a spreadsheet for root testing

Ultimately, this is a root finding problem, so, yes, you need to develop a spreadsheet you can work with for finding roots. If you are willing to go through the tedium of calling goal seek manually, your current spreadsheet will work for this. If you want to try to build off of one of my sheets, where you have limited mathematical literacy, the secant version (post #7) might be the easiest to adapt. You should only need to replace my composite polynomial function in B8:D8 and F8:H8 with your function. You could maybe:
1) Because this function cannot handle negative numbers, let's work in row 12, which defaults to a positive initial guess for x.
2) Enter 5 (your given value for D) in B12.
3) Enter 0.05 (your target percentage) in C12.
4) Enter your formula in D12 =$B12/(A12+ln(A12))-$C12 -- note the mix of relative and absolute references.
5) Select D12 and copy into H12.
6) Spreadsheet should iterate and find the root. Check the solution and see if it is correct.
7) "Copy" (however that works best for you) row 12 into your spreadsheet and make however many copies you need.

That's more "cookbook" programming that programming from a solid understanding of the algorithm, but maybe that is good enough for now. If you ever need to edit or adapt to changing circumstances, those edits may be difficult unless you spend some time now understanding exactly how the algorithm works.

If you really don't like any of those options, we also have many examples of using VBA to call Goal Seek and/or Solver in a loop to solve these kinds of problems:
Using goal seek: https://www.excelforum.com/excel-pro...-function.html
Using Solver: https://www.excelforum.com/excel-pro...lver-loop.html
I don't usually find this approach easier to program, but some users do find it easier.

At this point, you need to decide which approach is going to best use your time and effort.

It looks like you got the same root I did when I tested this -- 95.44149 in A12. This looks like the root (or at least a root), because the value for y in D12 is 0. Are you expecting a result other than 95.4?

I'm not sure where the .998 number is coming from (other than it looks like the same value as is in A11). When I open both of your files, A12 immediately resolves to 95.4... I cannot see why it resolves to 0.998 on your computer.

What happens if you "reset" (enter TRUE in B5) the algorithm, then enter FALSE in B5?

That's what Excel does with circular references when you have iteration disabled. I guess I was not clear in my tutorial, but any iterative process like this that uses circular references is only going to work if you enable iteration (Excel options -> calcultion options -> check the box/radio button that enables iteration).