mathematics power number raised obtain value real number formula example base function function power function alternative include prior statement default base log historically application music theory frequency ratio musical tones gives number octaves tones differ length representation number system number bits needed message information theory computer science count number steps needed search related areas frequently include design sports tournaments photography included standard mathematical functions mathematical software packages found find set operation value looking floating point value calculated efficiently history powers known antiquity instance appear elements props powers half structure perfect numbers power position ordered sequence powers basis credited publishing known table book contains tables corresponding powers reversing rows tables allow interpreted tables earlier 8th century mathematician credited precursor concept defined number times given number divided evenly definition gives rise function powers different giving order modern form applying number powers considered explicitly established application music theory long significant applications information theory computer science known work area published table seven accuracy definition properties function defined function power function strictly increasing function positive real numbers unique alternatively defined natural defined standard ways complex definition allows extended complex numbers following combine formula formula formula list identities mathematics number written function proposed especially application areas authors write listed credits suggestion use information theory computer science dates active written log prior statement default base function especially german scientific literature latin din standards recommend according standards instead reserved applications information theory number bits representation positive integral formula information theory definition self information information expressed corresponding making bit fundamental unit information natural alternative definitions natural important areas pure mathematics number theory mathematical analysis applications tree leaves height equality power tree complete tree number river system tributary streams family sets different sets elements union equality family power set partial cube dimension edges equality partial cube according independent set size precise size guaranteed known best bounds known size involve particular independent set size independent set size larger mathematical analysis gilbert shannon model random number times needs shuffle card deck cards distribution close uniformly random approximately calculation forms basis recommendation card decks seven times complexity frequently appears analysis frequent use number occur analysis based way branching problem initially choices solution iteration reduces number choices factor number needed select single choice integral idea analysis data structures example search size problem solved iteration roughly needed obtain problem size solved easily constant time similarly perfectly balanced search tree containing elements height running time usually expressed big expressions constant factors lower order terms different bases differ constant factor run time said run time base expressions log log important omitted appear time bound base omitted example equal running time log called examples running time log log average time comparison sort searching balanced search trees longest increasing occur time bounds divide conquer bit numbers time time occurrence running times explained reference master measure strongly different genes expressed sample biological material different rates expression gene compared ratio expression rates log ratio expression rates defined ratio rates allow convenient comparison expression rates doubled expression rate described log ratio expression rate described log ratio unchanged expression rate described log ratio zero instance data points obtained way coordinate axes intensity plot plot rotate scale log ratio music theory music theory interval difference tones determined ratio frequencies intervals coming rational number small perceived particularly simplest important intervals octave frequency ratio number octaves tones differ frequency ratio study tuning systems aspects music theory require finer tones helpful measure size interval finer octave frequency tones form rising sequence tones measure interval plus measure interval equal measure interval measure given cent divides octave equal intervals cents given tones frequencies number cents interval formula defined way instead sports scheduling competitive games sports involving players teams game match indicates number rounds necessary single elimination tournament required determine winner example tournament players requires rounds determine winner tournament teams requires rounds etc case players teams power rounded necessary round remaining competitors play example approximately rounds indicating tournament teams requires rounds teams sit round team sits second round number rounds necessary determine clear winner swiss system tournament photography photography exposure values measured terms light reaching film accordance law describing response human visual system light single stop exposure unit scale precisely exposure value photograph defined formula number measuring lens exposure number seconds exposure expressed stops express dynamic range light sensitive materials digital sensors calculation conversion bases easy way function use natural common log functions found scientific specific change base formula approximately formula rounding function rounding forms related formula formula definition extended defining formula extended way function related number leading bit unsigned representation formula interpreted zero based index significant bit input sense complement find set operation finds index significant bit hardware platforms include support finding number leading equivalent operations quickly find functions versions software library rounded plus general positive real number parts formula called characteristic reduces problem argument restricted range interval second step computing exists unique simply words formula normalized floating point numbers given floating point determined performing count leading operation result elementary division computing described follows start real number half open interval zero square repeatedly result lies interval let number needed chosen taking sides formula real number interval return step method result expressed following formula number required iteration formula special case step found zero finite sequence terminating point infinite series according ratio test term strictly previous practical use infinite series truncated reach approximate result series truncated term error result software library support function included standard mathematical functions default version function takes double precision arguments variants allow argument single precision long double argument function allowed negative number case result complex number
Binary logarithm