useful mathematical facts in this appendix we give several useful mathematical facts we begin with some combinatorial definitions and facts logarithms and exponents the logarithm function is defined as logb if bc the following identities hold for logarithms and exponents logb ac logb logb logb / logb logb logb ac logb logb (logc )logc blogc alogc (ba ) bac ba bc ba+ ba /bc ba- in additionwe have the followingproposition if and bthen log log log justificationit is enough to show that ab / we can write ab ab ab ( ) ( ( ) < the natural logarithm function ln loge xwhere is the value of the following progressione **