Question

Find the Algebraic answer, absolute uncertainty, and relative uncertainty of:

log [8.47(+/- 0.05)]

I'm having difficulty finding the uncertainty of logarithms and exponents, so showing work would be much appreciated!

Answer 1

The absolute uncertainty in a natural log (logarithms to base e, usually written as ln or loge) is equal to a ratio of the quantity uncertainty and to the quantity. Uncertainty in logarithms to other bases (such as common logs logarithms to base 10, written as log10 or simply log) is this absolute uncertainty adjusted by a factor (divided by 2.3 for common logs)
so (ln 8.47 +/- 0.05) would be ln (8.47) +/- 0.05/8.47
and
(log 8.47 +/- 0.05) would be log (8.47) +/- 0.05/(8.47*2.303)
(log 8.47 +/- 0.05) =   log (8.47) +/- 0.05/(8.47*2.303)
                                       =0.928 +/-2.9*10^-4
This gives absolute uncertainty.
Relative uncertainty = 2.9*10^-4 / (0.928) = 2.7*10^-4

source:

http://phys114115lab.capuphysics.ca/App%20A%20-%20uncertainties/appA%20propLogs.htm