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#1 2013-05-22 18:42:10

{7/3}
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Registered: 2013-02-11
Posts: 210

Help me prove this

Help me prove this:

for some constant a[i cannot use the fact this is ln(x)]

Last edited by {7/3} (2013-05-22 18:43:33)


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#2 2013-05-22 23:14:11

bob bundy
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Registered: 2010-06-20
Posts: 6,336

Re: Help me prove this

hi {7/3}

Are you wanting a proof from first principles?  I usually start with the derivative of a^x, then e^x, then reverse these for the log.

If you may assume d(e^x)/dx = e^x then it will only take a few lines.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#3 2013-05-22 23:50:21

{7/3}
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Registered: 2013-02-11
Posts: 210

Re: Help me prove this

Proof from first principles will be better


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#4 2013-05-23 01:26:37

bob bundy
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Registered: 2010-06-20
Posts: 6,336

Re: Help me prove this

This is how I do powers and logs:

For the function

at (0,1), the derivative is:

Even though I don't know what that is, it will have a value; let's say k.

Now the derivative at other points


So all graphs in the family have the property that the gradient function at x is a^x times the gradient at (1,0)

In the family there will be one value of a for which k = 1

Call that one a = e

then

so

Now suppose

Taking logs base e for the first expression:

Differentiating wrt x

which means we now know the value of k ... and

so  [still working on this last bit but I think I'll post before I lose it all]

No good.  I seem to be stuck here because if k - ln a this becomes ln x and I was trying hard to avoid that.  I seem to have gone too far and proved the log is base e.  I'll come back to it later after a think.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#5 2013-05-23 03:32:01

bob bundy
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Registered: 2010-06-20
Posts: 6,336

Re: Help me prove this

OR

I suspect there's a circular argument lurking here as power series probably depend on natural logs somewhere, but maybe it's ok.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#6 2013-05-23 04:10:32

anonimnystefy
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Registered: 2011-05-23
Posts: 15,507

Re: Help me prove this

Hi Bob

The Taylor series uses the derivatives, so you are still using the derivative of log there. Unless we get the Taylor series of log in a different manner.


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#7 2013-05-23 07:35:27

bob bundy
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Registered: 2010-06-20
Posts: 6,336

Re: Help me prove this

OR

This was posted by yeyui on

http://forums.xkcd.com/viewtopic.php?f=17&t=36281

I have kept the post but edited the variables to suit your problem.

First some change of variable magic:

Now apply this magic to the function of interest


So this function has the property f(ab)=f(a)+f(b) which means that it is some logarithm.

Now "just" evaluate it at any particular point to show that it is the right one.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#8 2013-05-23 08:07:33

anonimnystefy
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Registered: 2011-05-23
Posts: 15,507

Re: Help me prove this

I really like that proof. Really elegant.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#9 2013-05-23 14:27:51

{7/3}
Member
Registered: 2013-02-11
Posts: 210

Re: Help me prove this

That was an awesome proof,but i need one more favor,if f'(x)=f(x) and f(0)=1 than f(x)=a^x for some constant a,how do i prove this?


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#10 2013-05-23 18:40:18

bob bundy
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Registered: 2010-06-20
Posts: 6,336

Re: Help me prove this

There are a whole set of functions that differentiate to give themselves.  But they are all multiples of each other.

That is, if f'(x) = f(x) and g'(x) = g(x) then f = kg for some constant k.

Proof:

Consider

Show this is equal to zero which means that h(x) = constant.

I think that should enable you to do what you want.


Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#11 2013-05-23 23:30:24

{7/3}
Member
Registered: 2013-02-11
Posts: 210

Re: Help me prove this

Thanks


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