Maths.

I do have to admit most definitions are somewhat circular

Are you comfortable with logarithms in general? Just asking so I know how to answer best.

Del has said it all but I will try and help you further

Number Log base 10

1 0

10 1

100 2

1000 3

10000 4

Long ago we used logs extensively to perform (now simple) calculations

If you have to calculate X by Y

you looked up the logs of both numbers and ten add them together and looked up the anti-log

For example 10 x 100 == 1 + 2 = 3 == 1000.

0r for 2 x 5 (hoping that Del's 0.3010 is correct)

Log (2) = 0.3010

Log (5) = 0.6990

Total = 1 and the result is 10.

I know what is 1+1 and can help myself with the basics. I use "log" and "e" on a daily basis in some work related formulas but I do not know why\how\what they really are.

It's like a lot of stuff it is to some extent arbitary...we could use any base..and we use it because it is convenient.

I once had a guy spend ages in some psedo mathematical convoluted explanation of why we use hexadecimal...
When the real, simple, reason is that's it's just a way of expressing 4bits of binary as a single digit which is of more convenient than having to use 2 digits as you would with base 10 or Octal...

Del

Generally "e" is specifically associated with natural logarithm or log natural (ln). With that said... Ug!!!

Perhaps the following link will help some:

http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

The cool thing about the natural number "e" is that in the function: y = e^x

It is it's own derivative, that is the slope at any point found by y = e^x, is exactly the same (e^x)

Works the same the other way with the integral, the area under the graph of y = e^x, from (0 to x) is e^x

Whoops - that is not true about the integral/area (but it is regarding the derivative/ slope)

For example:

On the interval x = 0 ......... 1

Area under e^x = e¹ - e⁰ = 2.71828... - 1 = 1.71828....

eh,,wait a minute,, isn't the antiderivative of 1e = e^1 or e ?

In other words, weren't you right the first time?

Hello CG

Yes he was right first time. The indefinite integral of e^x = e^x. The fact that you can get any answer you like for the area under the curve (by selecting integration limits) is irrelevant.

Cheers........Codey

ok, now I'm confused again. I never did understand that whole derivative of e = e thing. Why isn't the derivative of e = 1 ?

The derivative of a constant is zero.

The derivative of e to the power of x (e^x) is (e^x)

"The derivative of a constant is zero."

Check. e is a constant, not a variable. My mistake.

Which is why "The derivative of e to the power of x (e^x) is (e^x)" puzzles me.

I guess it has something to do with e being an actual rate/slope and a limit. Kind of like one of those einstein-rosen bridge thingies between derivatives... I suppose I would have to graph it to see it.

So e is both the derivative and integral of e, but only in positive terms (growth). In negative terms (deterioration / decay), e is the derivative of e, but not the integral of e. Right or wrong?

e is not the derivative and integral of e ....... e^x is. See below:

In negative terms, this is not the same.

The derivative of y=e^-x is -e^-x

Codey and Guest, Thank You!

oops, looking back , I realize that the link I gave you dealt primarily with e but not natural logs. Here's another that better explains log natural. Just skip the paragraph on e if you read the previous link, but read the previous link first, otherwise it might not make as much sense.

http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/

A log is not an "are", it is a technique for doing very fast, accurate math in your head or with a table,

SO - get in the wayback machine with this knowledge and without your calculator and look at Hendrik's answer again:

log₁₀ of 1 is 0

log₁₀ of 2 is 0.3010

log₁₀ of 5 is 0.6990

log₁₀ of 10 is 1

100 = 2

1000 = 3

10000 = 4

Long ago we used logs extensively to perform (now simple) calculations

If you have to calculate X by Y

you looked up the logs of both numbers and ten add them together and looked up the anti-log

For example 10 x 100 == 1 + 2 = 3 == 1000.

So far so good?

e^iΠ + 1 = 0

(CR4 doesn't do the symbol for pi very well. π = 3.141592653654......... and i = √-1 )

Maths is all made up by people too lazy to do it empirically
Del

8 = 2³ (2x2x2) is fairly easy to remember.

Put another way, Log₂8 = 3 ('Log of 8 to base 2 equals 3')

'Log' is usually taken to mean 'base 10', so Log 8 = 0.903089987 , 8 = 10^0.903089987

Base 10 is a little more intuitive. 100 = 10² and 1000 = 10³ . When multiplied, 100 x 1000 = 100000 = 10²⁺³ = 10⁵ . So, you can see how to simplify multiplication by adding terms.

'Natural log', indicated by ln, uses base e (2.718281828). It can be used in an identical way to base 10 ('Log')

ln 8 = 2.0279441532 ie 8 = e ^2.0279441532 = 2.718281828 ^2.0279441532

logarithms will work using any number as base, but using other than 10 or e will get surprised looks from colleagues.

'e' arises from summing series (Bernoulli/compound interest in particular). When looking for patterns in data, it's sometime useful to plot the logarithmic values. Often this helps establish the mathematical relationship by giving a straight line graph. Nature doesn't multiply by 10's very much, but it likes to compound effects, much like Bernoulli's interest.

It's something like that - I was thinking aloud !