It is always preferable to use a calculator or look at tables. But, before that a few things first.
Note that #log1=0#, #log10=1#, #log100=2#, #log1000=3#, #log10000=4#. Hence taking log of such a number (i.e. greater than or equal to one), count the number of digits and say it is #n#, then its log will be #(n-1)# plus some thing. Here we have to find log of #2006# and as it has four digits, it will be #3.(.....)#. This is called characteristic.
However, if the first significant digit (from left) starts after decimal point, for example in #0.0002006#, it starts from fourth place after decimal, the characteristic will be #-4#.
Characteristic comes in the log before decimal point and what comes after the decimal point is called mantissa. Digits starting from first significant digit decide this. For example in #1#, #10#, #100#, #1000# and #10000#, it is just #1000# and hence mantissa is #0#. Mantissa is always positive and if characteristic is negative, one needs to add them like two real numbers.
One usually looks at first four digits and looks in tables. For number #456.7#, while characteristic is #2#, for mantissa, one looks at #45^(th)# row and then #6^(th)# column. The figure for this in table is #6590# and finally the #7^(th)# sub-column, which adds to this number and in this case it is #7# and hence mantissa for #4567# is #6597# and log of #456.7# will be #2.6597#.
As #2006# has four digits, hence characteristic is #3#, for mantissa look at #20^(th)# row #0# column (which shows #3010#) and #6^(th)# sub-column i.e. add #13# to it and hence #log2006=3.3023#
