# the exponential function

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what is so special about the number e and why can so many things be modeled by it? As in, I know why that the number π exists because it's the ratio of diameter to circumference of a circle, but e... just seems to pop out of nowhere!

just finished learning about capacitors, nuclear decay and chemical half lives, so this is quite important to my understanding of these concepts

tthnaks

just finished learning about capacitors, nuclear decay and chemical half lives, so this is quite important to my understanding of these concepts

tthnaks

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#2

Hi there,

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While you're waiting for an answer, did you know we have 300,000 study resources that could answer your question in TSR's Learn together section?

We have everything from Teacher Marked Essays to Mindmaps and Quizzes to help you with your work. Take a look around.

If you're stuck on how to get started, try creating some resources. It's free to do and can help breakdown tough topics into manageable chunks. Get creating now.

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#3

(Original post by

what is so special about the number e and why can so many things be modeled by it? As in, I know why that the number π exists because it's the ratio of diameter to circumference of a circle, but e... just seems to pop out of nowhere!

just finished learning about capacitors, nuclear decay and chemical half lives, so this is quite important to my understanding of these concepts

tthnaks

**genuinelydense**)what is so special about the number e and why can so many things be modeled by it? As in, I know why that the number π exists because it's the ratio of diameter to circumference of a circle, but e... just seems to pop out of nowhere!

just finished learning about capacitors, nuclear decay and chemical half lives, so this is quite important to my understanding of these concepts

tthnaks

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#4

**genuinelydense**)

what is so special about the number e and why can so many things be modeled by it? As in, I know why that the number π exists because it's the ratio of diameter to circumference of a circle, but e... just seems to pop out of nowhere!

just finished learning about capacitors, nuclear decay and chemical half lives, so this is quite important to my understanding of these concepts

tthnaks

Just as 2^20 is the same as 4^10, the base of any power function can be converted to a new base.

E.g. 2^20 = e^(ln(2)*20).

So any function like y = e^(kx), can be written in a form with a different base instead of e, like y = 10^(kx), but the values of k will differ.

So the question is, why is e chosen to be the base of so many equations, rather than 10, or 2, or any other number. The answer is because the gradient of the function y = e^x is also given by the function y = e^x. e is unique in this way, and therefore it allows calculations of gradients much easier using the chain rule, than what would be possible if the function was written in a different form like y = 10^(kx). The second form is still valid, but it is harder to take the derivative of this second base 10 form.

I don't know if you have covered derivatives and the chain rule yet, but when you learn it you will understand why e is chosen.

It may be of interest for you that e^x can also be expressed by this infinite function:

where the expression 3! simply means 3x2x1. Similarly, 4! means 4x3x2x1 and so on.

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#5

(Original post by

Anything that can be modelled with "e" can be modelled with any other base as well.

Just as 2^20 is the same as 4^10, the base of any power function can be converted to a new base.

E.g. 2^20 = e^(ln(2)*20).

So any function like y = e^(kx), can be written in a form with a different base instead of e, like y = 10^(kx), but the values of k will differ.

So the question is, why is e chosen to be the base of so many equations, rather than 10, or 2, or any other number. The answer is because the gradient of the function y = e^x is also given by the function y = e^x. e is unique in this way, and therefore it allows calculations of gradients much easier using the chain rule, than what would be possible if the function was written in a different form like y = 10^(kx). The second form is still valid, but it is harder to take the derivative of this second base 10 form.

I don't know if you have covered derivatives and the chain rule yet, but when you learn it you will understand why e is chosen.

**Doctor_Einstein**)Anything that can be modelled with "e" can be modelled with any other base as well.

Just as 2^20 is the same as 4^10, the base of any power function can be converted to a new base.

E.g. 2^20 = e^(ln(2)*20).

So any function like y = e^(kx), can be written in a form with a different base instead of e, like y = 10^(kx), but the values of k will differ.

So the question is, why is e chosen to be the base of so many equations, rather than 10, or 2, or any other number. The answer is because the gradient of the function y = e^x is also given by the function y = e^x. e is unique in this way, and therefore it allows calculations of gradients much easier using the chain rule, than what would be possible if the function was written in a different form like y = 10^(kx). The second form is still valid, but it is harder to take the derivative of this second base 10 form.

I don't know if you have covered derivatives and the chain rule yet, but when you learn it you will understand why e is chosen.

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#6

**genuinelydense**)

what is so special about the number e and why can so many things be modeled by it? As in, I know why that the number π exists because it's the ratio of diameter to circumference of a circle, but e... just seems to pop out of nowhere!

just finished learning about capacitors, nuclear decay and chemical half lives, so this is quite important to my understanding of these concepts

tthnaks

.

So

where Const. depends on the numerical value of . The fundamental definition of the number is then the value of such that Const. = 1 in this equation.

That is, the number is defined by the condition that

which rearanged becomes

.

Try on your calculator to get an approximation of . You should get 2.718.

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(Original post by

Using the property where is any positive number we get

.

So

where Const. depends on the numerical value of . The fundamental definition of the number is then the value of such that Const. = 1 in this equation.

That is, the number is defined by the condition that

which rearanged becomes

Try on your calculator to get an approximation of . You should get 2.718.

**steve44**)Using the property where is any positive number we get

.

So

where Const. depends on the numerical value of . The fundamental definition of the number is then the value of such that Const. = 1 in this equation.

That is, the number is defined by the condition that

which rearanged becomes

**.**Try on your calculator to get an approximation of . You should get 2.718.

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#9

(Original post by

thanks, this is a pretty clear explanation of what e is. How do you rearrange that first expression to get the one in bold though?

**genuinelydense**)thanks, this is a pretty clear explanation of what e is. How do you rearrange that first expression to get the one in bold though?

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