exponents xa xb xa+ xa xa- xb (xa ) xab xn xn xn  + logarithms in computer scienceall logarithms are to the base unless specified otherwise definition xa if and only if logx several convenient equalities follow from this definition theorem loga logc logc abc  proof let logc by logc aand loga thenby the definition of logarithmscx bcy aand az combining these three equalities yields cx (cy ) thereforex yzwhich implies /yproving the theorem theorem log ab log log bab proof let log ay log band log ab thenassuming the default base of band ab combining the last three equalities yields ab thereforex zwhich proves the theorem some other useful formulaswhich can all be derived in similar mannerfollow log / log log log(ab log log log log for all log , log , ,