so... i didn't really know where to ask so i came here
im in the first year of my computer science degree, and obviously learned about E.
the natural base of the logarithm.
i was just wondering, what is so special about E that it is the "natural base if the log"
i know that (E^x)' is E^x, but what that have to do with the log function?
why couldn't it be 7, or 1.0001, or anything else?
e is called the natural base of the log becuase it is that.
when you differentiate a^x, you get the deritave as a^x*ln(a) where ln is the natural log. e is the only number so that ln(e) = 1.
This means whenever you differentiate, or integrate e^x, it remains e^x.
So e^x is the inverse of the ln(x) function because ln(e) = 1 so ln(e^x) = xln(e) = x
Yes, the only special thing about e is that it remains unchanged under differentiation - or the slope of the curve equals the position along the x-axis at any point. The value for e is derived from this property (quite easy to do). This is a very useful property as there are many natural laws where the rate of change of some value equals the amount of that value.
yes i am aware of the definition of e.
but then again, why is e the natural base of the log?
so far i dont know about any big connection between the e^x derivative to anything about logarithms.
The natural logarithm just tells you what power to raise e by to get some other known value. So it's just the opposite of e^x. Since e^x is useful for many things, the natural log is also useful for working out its inverse. There's nothing else special about the natural logarithm compared to logarithms of other bases.
לוגריתם טבעי הוא לוגריתם שבסיסו הוא הקבוע המתמטי" e, שהוא מספר טרנסצנדנטי המתחיל בספרות 2.718281828. המונח הוכנס לשימוש לראשונה בשנת 1668 על ידי ניקולאוס מרקטור."
To be honest, I just found that more confusing http://en.wikipedia.org/wiki/Natural_logarithm
לוגריתם טבעי הוא לוגריתם שבסיסו הוא הקבוע המתמטי" e, שהוא מספר טרנסצנדנטי המתחיל בספרות 2.718281828. המונח הוכנס לשימוש לראשונה בשנת 1668 על ידי ניקולאוס מרקטור."
To be honest, I just found that more confusing http://en.wikipedia.org/wiki/Natural_logarithm
I meant after that:
הלוגריתם לפי בסיס e קרוי "טבעי" משתי סיבות:
קל להגדירו כאינטגרל פשוט או כטור טיילור עם מקדמים רציונליים - מאפיין שנדיר בלוגריתמים לפי בסיס אחר.
פונקציית האקספוננט הטבעי- מופיעה בתחומים שונים במתמטיקה הרבה יותר מהפונקציה , ובהתאם גם הפונקציה ההפוכה שהיא הלוגריתם הטבעי, יותר נפוצה מהלוגריתם עם בסיס עשר
Using the equation above, you could derive the number e from geometry and trigonometry by:
\( \left\{ \begin{matrix}
\cos \left( A+B \right)=\cos \left( A \right)\cos \left( B \right)-\sin \left( A \right)\sin \left( B \right) \\
\sin \left( A+B \right)=\sin \left( A \right)\cos \left( B \right)+\cos \left( A \right)\sin \left( B \right) \\
\end{matrix} \right. \)
[Link]
and it makes up hyperboloic functions (which have many applications):
\( \cosh \left( x \right)=\frac{e^{x}+e^{-x}}{2},\text{ }\sinh \left( x \right)=\frac{e^{x}-e^{-x}}{2} \)
And, even though these is related to the fact that \( \frac{d}{dx}\left[ e^{x} \right]=e^{x}\ln \left( e \right)=e^{x} \),
In the definition of the Laplace Transform:
\( \int\limits_{0}^{\infty }{e^{-st}f\left( t \right)dt} \)
It's used in logarithmic differentiation:
\( \frac{d}{du}\left[ a\left( u \right)^{b\left( u \right)} \right]=\left( a\left( u \right)^{b\left( u \right)} \right)\left( \frac{d}{du}\left[ b\left( u \right)\ln \left( a\left( u \right) \right) \right] \right) \)
It's used in L'Hospital's rule (evaluating complicated limits).
It makes up solutions for a whole lot of differential equations.
One special function it makes up is the erfi (error) function, which has many applications.
EDIT:
Another special function is the gamma function:
\( \Gamma \left( z \right)=\int\limits_{0}^{\infty }{t^{z-1}e^{-t}dt} \)
This is a neat function because you can not only compute factorials of positive integers, but of other real numbers.
\( \Gamma \left( z+1 \right)=z! \). So for example, \( \pi !\approx 7.18808273 \)
And, based on the fact that we can represent elementary functions which are continuously differentiable as a Taylor series, we can express other functions with the gamma function:
\( f\left( x \right)=\sum\limits_{n=0}^{\infty }{\frac{f^{\left( n \right)}\left( a \right)\left( x-a \right)^{n}}{n!}} \)
thanks for the answer cmowla, but the thing i dont get is not why e have sooooo many important applications, but it got the title" the natural base of the logarithm"
what is so "natural" about ln that does not exist for other log functions.
thanks for the answer cmowla, but the thing i dont get is not why e have sooooo many important applications, but it got the title" the natural base of the logarithm"
what is so "natural" about ln that does not exist for other log functions.
thanks for the answer cmowla, but the thing i dont get is not why e have sooooo many important applications, but it got the title" the natural base of the logarithm"
what is so "natural" about ln that does not exist for other log functions.
I'm not an expert on this topic, but here is my stab at an answer. The natural logarithm of a number x is defined as the area from 1 to x under the curve:
\( f(t) = \frac{1}{t} \)
Written more precisely this is:
\( ln(x) = \int_1^x \frac{dt}{t} \)
I don't have a proof of this already sketched out or anything, but the basic argument is that it can be shown that:
\( ln(e) = \int_1^e \frac{dt}{t} = 1 \)
And since \( ln(e)=1 \) then e is the base of the natural logarithm.
Hope this helps, and I'll see if I can find a proof of the claim that ln(e)=1 using the integral definition of natural logarithm.
Historically, the original definition of the natural log function is as Chris posted above.
There are many definitions and applications of the number e. The same goes for the term "natural log." Different people will give you different a reason for the natural log being natural.
It's even worse when you're lecturer can't make his mind up on which one to use :/
(Out of interest Ben, are you studying A-level maths now? Considering you've differentiated x^(x^(x^(x))) which is something doable with the tools A-level maths gives you and a little insight.)
I'm not an expert on this topic, but here is my stab at an answer. The natural logarithm of a number x is defined as the area from 1 to x under the curve:
\( f(t) = \frac{1}{t} \)
Written more precisely this is:
\( ln(x) = \int_1^x \frac{dt}{t} \)
I don't have a proof of this already sketched out or anything, but the basic argument is that it can be shown that:
\( ln(e) = \int_1^e \frac{dt}{t} = 1 \)
And since \( ln(e)=1 \) then e is the base of the natural logarithm.
Hope this helps, and I'll see if I can find a proof of the claim that ln(e)=1 using the integral definition of natural logarithm.
Eulerftw. But, I hate \( \theta \) D:
