Here #log# means logarthim to the base #10#. When we take logarithm of a number, there are two parts of it; one part is characteristic and other mantissa.
While characteristic is the integral part and mantissa is the fractional or decimal part. For example #log500=2.6990#. Here #2# is characteristic and #0.6990# is mantissa.
While characteristic can be any integer, mantissa cannot be a negative number and is always positive. For example we write #-2.3010# as #-3+0.6990# i.e. a sum of an integer and a positive proper fraction and here for #-2.3010#, charcteristic is #-3# and mantissa is #0.6990#.
Characteristic depends on the place from where the number starts. For example, for a three digit number like #523# it is #2#, for a six digit number #743892# it is #5#. If we have a number #8.375#, characteristic is #0#.
What about numbers less than #1#, such as #0.893# or #0.00893# or #0.00000893#. In such cases characteristic is negative and depends on the place from where the number starts. For #0.893# characteristic is #-1#; for #0.00893# it is #-3# and for #0.00000893# it is #-7#.
Mantissa on the other hand is independent of the position of the decimal point in the number and just depends on first four digits, excluding #0's# on the left and is given in the logarithmic tables.
Hence as in #log0.000678#, number starts from fourth place after decimal, characteristic is #-4# and tables give mantissa as #83123# (they are easily available on web) and hence
#log0.000678=-4+0.83123# and is also written as #bar(4).83123#.
Scientific calculators give it as #-3.168773#
