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.