ab ab ab / ( ) ( ) / <( ) / / the natural logarithm function lnx log xwhere is the value of the following progressione / / / **in additionex / / / **ln( xx / / / **there are number of useful inequalities relating to these functions (which derive from these definitionsproposition if / <ln( < proposition for <ex < / proposition for any two positive real numbers and ninteger functions and relations the "floorand "ceilingfunctions are defined respectively as follows the largest integer less than or equal to the smallest integer greater than or equal to the modulo operator is defined for integers > and as