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In most undergraduate textbooks for chemistry and physics, I'm able to find rules to handle the case where the exponent is exactly 2 -- but I'm not able to find rules for the case where the exponent also has significant figures.

Only some college websites I have visited even have rules for logarithms and exponentiation at all; but those who discuss Napieran logarithms (log base e) generally include a disclaimer saying that the "actual" rules are more complicated -- and then sadly cite no sources for further study.

For example: laney . edu /wp/cheli-fossum/files/2011/01/Significant-Figure-Rules-for-los.pdf

I don't see a problem with actually following the rules as given (I'll give a worked out example), but I am concerned that there are supposedly places that the rules break down for Naperian logarithms as opposed to log base 10; Does anyone know the more complete rules for napierian logarithms or why there is an issue at all? I'm forced to use ln() (base e) on the computer system I do analysis on -- and don't want mistakes creeping in if I can avoid them.

As a worked example; I think I'll compute: 3.05**2.01

Please note, I am only going report the s.f. of each step -- but the non reported (insignificant) figures are still used in subsequent steps to improve rounding accuracy.

To solve it, I'll convert it to a logarithm and exponent expression.

3.05**2.01 = exp( ln(3.05) * 2.01 )

ln(3.05) becomes 1.115 because the 3 s.f. in the original number becomes a mantissa of 3 places (.115) in the logarithm. So the result of this ln(), is to produce a number with a total of 4 s.f.

Next:

1.115 * 2.01 becomes 2.24 since four s.f times three s.f, is three s.f.

exp(2.24) becomes 9.4, eg: a mantissa of 2 digits, becomes two s.f. in the result.

so: 3.05**2.01 ~= 9.4 (according to laney.edu rules extrapolated naiively.)

As an error analysis, I followed laney.edu's example and inspected neighboring calculation values.

3.04**2.00 = 9.2416...

3.05**2.01 = 9.4068...

3.06**2.02 = 9.5754...

eg: error ~=+-0.17

So, it does appear that there are in fact only 2 s.f. in the final result, as the error term varies in the first decimal place -- even though there were 3 s.f. in both the values passed into the power equation and I therefore *lost* 1 s.f. due to calculation.

And, for a compare and contrast example; If I redo the example with an infinitely precise 2 as the exponent which in theory should be a 3 s.f. result:

3.04 ** 2 = 9.2416

3.05 ** 2 = 9.3025

3.06 ** 2 = 9.3636

eg: error ~= +-0.061

So it does appear there are about 3 s.f. in the final result, as the error term varies in the second decimal place -- although one might argue that the first digit did change in the example result of 9.24... vs 9.30...

So -- I don't really see any problem with the results of my random example, but I'd like to know if anyone knows the more complete rules that laney.edu and a few other colleges refer to but don't list out ?