Zyra's front page //// Concepts //// Science //// Mathematics //// Site Index

2.7182818284590452...

Log e Explained

Log e is the Natural Logarithm - logarithms which instead of using base 10 use the number e which is 2.718281828459045235360... an interesting number.

First, what are base10 "common logarithms"? See Logarithms Explained. But at least they are based on a sensible number, TEN. Why would someone want to use a different base, such as 2 or 3? And why ever use a really odd irrational number such as the never-ending decimal e which is 2.718281828... , a number of the ilk of PI?

The first thing is that for logarithms you can in theory use ANY base. For example, supposing a jug was said to be a quart, it would be twice as big as a pint. If it was a gallon it would be twice twice twice a pint, (8 pints). Don't worry about the oldfashioned units. The key point is that the word "twice" was used three times, but adding up the words didn't add up the meanings, it multiplied them! This shows logarithms of base 2 being used. And the conjuring trick of logarithms, in which adding up logs makes the actual numbers multiply, works, for any base. The only reason base10 was used as a base for Common Logarithms was because it seemed reasonable at the time, like the way the counting number base was chosen to be 10.

Log e is surely a different matter though, as it's not an integer?

The reason why e is what it is and why it is chosen as a logarithm base is because it is the only logarithm base which has a rate of change the same as the thing which is changing. If something was getting bigger at a rate proportional to how big it was already it would be termed "exponential growth". When you work out equations with these types of changes and rates of change (known as differential equations), it's very handy to have this thing e which changes at a rate the same as itself.

To put it another way, the differential of ex = ex , or the slope of a graph of ex is ex , whereas for other numbers, that's not so.

The number e also appears in a variety of other places in maths, for example, e is what you get if you add together the reciprocals of all the factorials (0!+1!+2!+3!+4!...etc). Also, in the formula (1 + 1/x)x , where x is a number, the larger x gets, the nearer the answer comes to e. (You can try this experimentally on a scientific calculator!)

Was that explained well? Or was it as clear as mud? Can you explain it better? Perhaps you'd like to help improve this page.

Do you like this site? Is this sort of stuff useful? If you've found this page by a search, here's some more about this site: [response]

Logarithms have other applications, for example deciBels are measured on a logarithmic scale. Another example of a logarithmic scale is the Richter Scale which is used for measuring earthquakes.

Also see COMMON LOGARITHMS (base 10)

Care to see Natural Logarithm log-e to a few more places...

2.7182818284590452353602874713526624977572470936999595749669676277240766303535 47594571382178525166427427466391932003059921817413596629043572900334295260595630 73813232862794349076323382988075319525101901157383418793070215408914993488416750 92447614606680822648001684774118537423454424371075390777449920695517027618386062 61331384583000752044933826560297606737113200709328709127443747047230696977209310 14169283681902551510865746377211125238978442505695369677078544996996794686445490 59879316368892300987931277361782154249992295763514822082698951936680331825288693 98496465105820939239829488793320362509443117301238197068416140397019837679320683 28237646480429531180232878250981945581530175671736133206981125099618188159304169 03515988885193458072738667385894228792284998920868058257492796104841984443634632 44968487560233624827041978623209002160990235304369941849146314093431738143640546 25315209618369088870701676839642437814059271456354906130310720851038375051011574 77041718986106873969655212671546889570350354021234078498193343210681701210056278 80235193033224745015853904730419957777093503660416997329725088687696640355570716 22684471625607988265178713419512466520103059212366771943252786753985589448969709 64097545918569563802363701621120477427228364896134225164450781824423529486363721 41740238893441247963574370263755294448337998016125492278509257782562092622648326...

see, it's like PI, goes on forever. If you'd like it to a million places, you can have it downloaded free at Project Gutenberg