ITT: Teach Arsal Basic Math

[Crow]

Continued from the shoutbox.

Define exponential function for me (e^x) and also ln.

I know e is a constant which is a derivative of itself.
I know ln is natural log with the base e.
I want to know HOW and WHY this is.
If anyone can, please explain this.

Other definitions you might like to take a shot at:

- Linear/Non-linear.
- Homogeneous/Non-homogeneous
- Differentiation/Integration (in its purest form, also explain it's genesis and how it came to be)

No dumb replies allowed. If you cannot contribute (even links work, honestly), don't spam the thread.

 

[mzain90]

okay here i go trying to reacall a 3 yer old add math lecture.
e or the exponention function is just a contant, the constant equal to ~ 2.718, the function is defined as...

now speaking as an analogue, e^x has the same properties as 2^x, the graph looks the same if you adjust the scale. the only thing sepcial abt this function is that its dervitive is its self, so basically on a graph of e^x, the gradient at any point x can be found by simple putting the value of x in the original function.....
cont.....

---------- Post added at 09:12 PM ---------- Previous post was at 09:04 PM ----------

now coming to log, log is basically used to approximate powers and verry large, or very small numbers.

basically if y=2^x, x=log with base 2 and the other thing Y.

the log in your scientific calculators is log to base 10,

now note if 10=10^x, x would be 1 ryt.
so when log 10 (base is 10 in this context) the answer would be one.
similarly 100=10^x, would give x=2
and log 100 would give 2


now 10 is a constant, so is e, so some nutcase decided it would be fun to tourture innocent souls by combining one semi confusing thing with another semi confusing thing to make something.

moving on....
so the crackpot made an expression og log with base e, its know as natural logarithm or ln

using the examples above we now that ln e would simple be 1

and hence ln e^x would simply be x,
and e^ln x, is also x yea its a really twisted function.


we use ln and e for various reasons, at add maths level we used e to cancel out ln, and ln to cancel out e, which is really a Abuses will lead to ban cause they made the functions up in the first place.

---------- Post added at 09:14 PM ---------- Previous post was at 09:12 PM ----------

a linear function is one when plotted on an x/y plane (graph) would give a straight line, i.e y=2x, y=2x+ any constant(e maybe)

non liner is any function which is NOT linear, rocket science aint it

---------- Post added at 09:15 PM ---------- Previous post was at 09:14 PM ----------



homogenity...

im going to over simplify it .
in an equation where x^2 +y^2 =xy

this equation would be homogenous because the the units would add up, basically suppose x and y have any unit say cm (centimeter) or a(any unit )
it would be homogenous because the units would become a^2 + a^2 =a^2
all the equations in physics etc are homogenous, eg

e=mc^2
m=Kg
c=speed of light=m/s^2
so energy is Kg.m^2/c^4


now using the formulae of potential energy, e=mgh, e would again add up with the same units.


homogenity of equations is pretty simple till is come into the context of homogenous differential equations, and finding out weather a differential equation is homogenous or not...that gets a bit more complex and i cant really explain over a forum.



oh boy differentiation and integration...
cont...

 

[Crow]

log and ln are starting to make a lot of sense to me. Thanks!

Could you elaborate on e just a little more?
How was the constant value determined?
Why was it determined?
Where does it exist in nature?

Thanks in advance!

 

[mzain90]

since you said basic math above, differentiation of a function is basically decomposing a function, and integration is adding it up,

siply when you differeciate a funtion, you get an equation of a tangent at that point, so you took a equation of a curve, and fro it, you took just one point and found out abt that individual point.


integration does the reverse, it adds up all the point of an equation,
so integration a funtion like y=x, it add up the values for y, at x=all the point within the limits.

its a handy way of finding the area under the graph.


although we use differenciation an integration to find out and solve the equation of electromegnetic waves travelling through air...and belive me you dont want ME to explain that

p.s, i got an A at 95% in add maths, Maths, and A level maths(consistent ryt, ), but nearly flunked ordinary and partial differential eqs in uni

---------- Post added at 09:51 PM ---------- Previous post was at 09:44 PM ----------

okay the constant value was determine by the equation i mentioned

the function is
1+1/1! + 1/2! + 1/3! + 1/4! ...till 1/infinity (basically x=1 at e^x)

=> 1+1+0.5+0.166+0.041...
ntice that the term after that only make subsequently smaller changes..after the first few terms we can find out the value ~2.7

---------- Post added at 09:55 PM ---------- Previous post was at 09:51 PM ----------

umm here is the thing, when you solve differential equation sometimes you get equtaion that you cant normally solve, so you make a series with unknowns, differnciate that and find the result of that differential, which also then comes in series...so this maybe how the e was made, i dont know for sure...google is your friend mate.

where does it exist....in the twisted minds of ppl who excel in academics :S

 

[Benighted]

Thanks, Mzain, your post did clear some things for me as well.

Linear functions like, Mr. mzain explained, are functions that give a straight line when they're plotted on a graph. I think the exponent of each term in a Linear equation is never larger than 1 (Could someone please confirm this?).

Differentiation, although I don't know its definition, I think Newton, Leibniz, Fermat etc. developed it for determining maxima and minima of things, and for determining tangents of curves and positions of planets.

I don't know about e either. Isn't 4^2 = e^(4ln2)?

Meh, I'm sure you already know the close-to-none information I've posted here, Arsal. lol. Sorry, man. Wish I could be of any help.

Looking forward to more valuable posts in this thread which can benefit my enlightenment while we're at it.

 

[Crow]

Thanks again mzain90, and VC!

I have some more questions (dang! lol) but I'm too tired to articulate. I'll come back to this in the morning.

 

[mzain90]

Differentiation, although I don't know its definition, I think Newton, Leibniz, Fermat etc. developed it for determining maxima and minima of things, and for determining tangents of curves and positions of planets.

+1, and yes linear functions are always of the order one...thats the proper definition, the graph one is good for concept, although by adjusting the scale you can plot non liner equations as a straight line too...the things thing they make us do...

Isn't 4^2 = e^(4ln2)?

lol no,
i know the answer matches but heres why

e^(4ln2)/* there is a propers of log of any base that if log 16, or log 2^4 it can be written as 4log 2; basically power can be separated, same applies no natural log.


so 4ln 2is basically ln 2^4
e^ln cancels out and what ever was the argument in ln remains.

so you get 2^4; which is mathematically equal to 4^2...you just got lucky there


you know next time there is a thread for wtb a level tutor, i think i'll reply (lmao)