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
