Your various equations appear to relate to compounded growth in various contexts. Instead of trying to pick "the right one", use themeaningto find what is helpful.

In general, compounded growth followsthe following formula:

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...where "A" is the ending amount, "P" is the initial amount ("principal", in a monetary context), "r" is the rate of growth (expressed as a decimal), "n" is the number of compoundings per growth period (usually the number of compoundings per year), and "t" is the number of periods (usually years).

In your case, you know that, whatever P you start with, you'll have A = 2P in t = 4 days. Let's plug that in:

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If we let n = 1, so there is one "compounding" per "period" (because this will simplify our work), then we get:

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Divide through by P to get:

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There are various ways to proceed from here. One would be to take the fourth root of each side, and then subtract the 1 to isolate r:

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Now use the same formula, but plug in the value you have for r. Set P equal to 1, set A = 10,000, and solve:

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Use logsto find the value for t.

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