I understand what you are saying, believe me. I am trying to keep it simple because the objective is for the child to walk away with a useful non-scary answer that gives them a sense of proportion with which they can approach thinking about these things.

Anyone who has tried to teach a child math is familiar with just how hard it can to have them understand seemingly simple concepts. Simple example unrelated to powers/logs/roots. It took me about half an hour to explain how you can shift a parabola right and left by simply adding or subtracting a constant from x in the simplest form y = x^2. The fact that it moves in a direction opposite the sign caused even more confusion. It took telling the story in five different ways before the "aha!" moment happened.

The relationship between exponentiation and logarithms is another one that gets fun once things are not nice and even. Exponentiation is sequential multiplication and logs sequential division. Sounds good, until you can't multiply or divide by the base any more.

I find it interesting that in all of my searching I have not found a simple approach to explaining these things to children so they can build a tangible sense of what's in front of them.

That said, if the kid understands coding, yes, you can use programs to have them explore how things might work, create solutions, understand errors, estimation, etc. More the reasons to perhaps teach coding and math in parallel and to the same level of importance in schools.