Here loglog means logarthim to the base 1010. 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=2.6990log500=2.6990. Here 22 is characteristic and 0.69900.6990 is mantissa.
While characteristic can be any integer, mantissa cannot be a negative number and is always positive. For example we write -2.3010−2.3010 as -3+0.6990−3+0.6990 i.e. a sum of an integer and a positive proper fraction and here for -2.3010−2.3010, charcteristic is -3−3 and mantissa is 0.69900.6990.
Characteristic depends on the place from where the number starts. For example, for a three digit number like 523523 it is 22, for a six digit number 743892743892 it is 55. If we have a number 8.3758.375, characteristic is 00.
What about numbers less than 11, such as 0.8930.893 or 0.008930.00893 or 0.000008930.00000893. In such cases characteristic is negative and depends on the place from where the number starts. For 0.8930.893 characteristic is -1−1; for 0.008930.00893 it is -3−3 and for 0.000008930.00000893 it is -7−7.
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
