Definition of e / Changing base question

1. Nov 25, 2014

RJLiberator

1. The problem statement, all variables and given/known data
Here's the problem:

Mod note: the last line is extraneous, and unrelated to this problem.

2. Relevant equations

Definition of e (1-1/n)^n, etc.

3. The attempt at a solution

My question is this:
a) When do you know to use e in a situation like this? Is it when you have exponents in such a situation?
b) Is this the definition of e? Or utilizing it somehow?
c) If there is any sites/information that you can give me on this procedure, I would be grateful. In my calc studies this must have been skipped over and now it is being applied as if it has been taught to me already.

Last edited by a moderator: Nov 25, 2014
2. Nov 25, 2014

Staff: Mentor

Where did the last line come from? The value of this limit is zero, and is totally unrelated to the problem of finding the derivative of $2^{x + 1}$.
An exponential function using an arbitrary base can be rewritten as an exponential function use e as the base. The basic idea is that $a = e^{ln(a)}$, so $a^x = (e^{ln(a)})^x = e^{xln(a)}$

3. Nov 25, 2014

RJLiberator

Mark44, sorry for the confusion, please exclude the last line. This was a snapshot of a calculation for a longer problem. I only had a vague idea of what was happening in this part and was looking for clarification :).

Thank you for your definition. I had that idea in my mind, but needed it laid out in front of me as you did. This helped me understand when to use e in a limit problem such as above. :)

Cheers

4. Nov 25, 2014

Staff: Mentor

But the problem you posted is not a limit problem (unless you invoke the underlying definition of the derivative). I think it would be clearer to describe it as a derivative problem.

5. Nov 25, 2014

RJLiberator

Ah yes, this was a limit problem applying l'hospitals rule so it became a derivative problem inside a limit problem. :)

Indeed, it should have been labeled as a derivative problem according to my issue.

6. Nov 25, 2014

Ray Vickson

You wrote
$$2^{x+1} = e^{\ln(2)(x+1)}\\ = \frac{d}{dx} \left( e^{\ln(2)(x+1)} \right)$$
Do you see why this statement is wrong?

7. Nov 25, 2014

Fredrik

Staff Emeritus
I think the equality on the second line was meant to continue the calculation on the line before the first line. The grayed out line is only the explanation for why the equality holds.

8. Nov 25, 2014

Fredrik

Staff Emeritus
a) You're supposed to prove that $\frac{d}{dx}2^{x+1}=2^{x+1}\ln x$ for all x. If you're allowed to use the formula $\frac{d}{dx}a^x =a^x\log_a x$, then the proof is a trivial application of that formula and the chain rule. If you're not allowed to use it, then you have to rewrite $2^{x+1}$ as something that you know how to deal with, and apparently you know how to deal with $e^{f(x)}$ when f is a differentiable function, so you write $2^x=e^{\ln 2^x}=e^{x\ln 2}$.

b) If your book defined the exponential function before the logarithm, then it probably defined the logarithm as the inverse of the exponential function. So $e^{\ln x}=x$ and $\ln e^x=x$ follow immediately from the definition of the logarithm.

c) The $x=e^{\ln x}$ rewrite is very common, so it should be mentioned in any book on calculus. It's basically just the definition of the logarithm, so there isn't much to say about it, but you should be able to find examples of high to use it in any book on calculus that includes some solved exercises.

9. Nov 25, 2014

RJLiberator

Excellent material Fredrik. This will help me out, surely.

10. Nov 25, 2014

Ray Vickson

Yes, I know that, but I was hoping the OP would see where the writing error lies, and promise to do better in future.