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C3 logs and exponentials

What's so special about y=exy=e^x???

I cant remember but my teacher said something about the gradient equalling the same y value n the y axis or something... o.o
Reply 1
Avatar for Reda2
Reda2
14
Original post by bigdonger
What's so special about y=exy=e^x???

I cant remember but my teacher said something about the gradient equalling the same y value n the y axis or something... o.o


Do ln of both sides you get lny = x
Differentiate or integrate y with respect to x and you get e^x again.
Reply 3
Avatar for JLegion
JLegion
2
Original post by bigdonger
What's so special about y=exy=e^x???

I cant remember but my teacher said something about the gradient equalling the same y value n the y axis or something... o.o


The gradient of y = e^x is equal to e^x.
When you differentiate y = e^x you get dy/dx = e^x... This is not true for y = e^2x (or any other function of x), however. You will learn this once you cover the differentiation chapter in C3 :smile:
Reply 4
Avatar for bigdonger
bigdonger
OP
2
Original post by Reda2
Do ln of both sides you get lny = x

ok... so what does that mean? ...
oh i see ok but is that's all that's special about y=e^x?
Original post by Wunderbarr
Differentiate or integrate y with respect to x and you get e^x again.

yup teacher told me that
Original post by JLegion
The gradient of y = e^x is equal to e^x.
When you differentiate y = e^x you get dy/dx = e^x... This is not true for y = e^2x, however. You will learn this once you cover the differentiation chapter in C3 :smile:

oh? is there proof for this? how does one know?
Reply 5
Avatar for JLegion
JLegion
2
Original post by bigdonger
oh? is there proof for this? how does one know?


You will learn this in the C3 differentiation chapter.

I believe the proof for this is beyond the A-Level syllabus :tongue:
(edited 7 years ago)
Original post by bigdonger
ok... so what does that mean? ...
oh i see ok but is that's all that's special about y=e^x?

yup teacher told me that

oh? is there proof for this? how does one know?


y = e^x
lny = lne^x
lny = x
(Implicit diff wrt x)
(1/y)(dy/dx) = 1
dy/dx = y
dy/dx = e^x

combine that with: https://www.khanacademy.org/math/differential-calculus/taking-derivatives/der-common-functions/v/proof-d-dx-ln-x-1-x
(edited 7 years ago)
Reply 7
Avatar for bigdonger
bigdonger
OP
2
Original post by JLegion
You will learn this in the C3 differentiation chapter.

I believe the proof for this is beyond the A-Level syllabus :tongue:

ah ok
Original post by XOR_
y = e^x
lny = lne^x
lny = x
(Implicit diff wrt x)
(1/y)(dy/dx) = 1
dy/dx = y
dy/dx = e^x

combine that with: https://www.khanacademy.org/math/differential-calculus/taking-derivatives/der-common-functions/v/proof-d-dx-ln-x-1-x


probably need to earn that bit, i understand the rest though.....
Reply 8
Avatar for B_9710
B_9710
18
e is a special number because it is the base of the natural logarithm. It is a special number and comes up a lot in growth and decay. It is defined in many ways but I think the most natural way is that it can be defined as
e=lim⁡n→∞(1+1n)n \displaystyle e= \lim_{n\to\infty } \left (1+\frac{1}{n} \right )^n .

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