Standard Deviations of Logarithms and Antilogarithms

The last two entries in Table 6-4 show that for y = log a

          (1.1)

and for y 5 antilog a

           (1.2)

As shown, the absolute standard deviation of the logarithm of a number is determined by the relative standard deviation of the number; conversely, the relative standard deviation of the antilogarithm of a number is determined by the absolute standard deviation of the number. Example 1.1 illustrates these calculations.

Example 1.1

Example 1.1c demonstrates that a large absolute error is associated with the antilogarithm of a number with few digits beyond the decimal point. This large uncertainty is due to the fact that the numbers to the left of the decimal (the characteristic) serve only to locate the decimal point. The large error in the antilogarithm results from the relatively large uncertainty in the mantissa of the number (that is, 0.4 ± 0.3).