# Derivative of log/exponential functions

• agenttiny200
In summary, the derivative of the function (2x)/(ex) is (ln2-1)(2/e)^x. The confusion about the log(2/e) term was due to the use of log instead of ln, as log(x) is the natural logarithm. Wolfram Alpha is a helpful tool for mathematical calculations.
agenttiny200

## Homework Statement

Find the derivative of the following function:
(2x)/(ex)

## Homework Equations

f'(ax) = ax . lna
f(ex) = ex

I already know the answer, but I don't understand how it comes to it. why does the log(2/e) (from below, 'The attempt at a solution' step 3) appear? and why does it turn into {log(2)-1}? from my understanding, it should turn into log(2) - log(e-1)

## The Attempt at a Solution

f(x) = (2x)/(ex)
f(x) = (2/e)x
f'(x) = {(2/e)x} . {log(2/e)}
f'(x) = {(2/e)x} . {log(2)-1}

No, the answer is ok. log(a/b)=log(a)-log(b) so log(2/e)=log(2)-log(e).

Dick said:
No, the answer is ok. log(a/b)=log(a)-log(b) so log(2/e)=log(2)-log(e).

But it is exactly with this that I am getting stuck, why does log(e) turn into 1? isn't it equal to 0.43429...?

agenttiny200 said:
why does the log(2/e) (from below, 'The attempt at a solution' step 3) appear?

agenttiny200 said:
But it is exactly with this that I am getting stuck, why does log(e) turn into 1? isn't it equal to 0.43429...?

So you got it? log to the base 10 of e is 0.43429... But that's not what you want here, right?

well, to explain it better, I used this site:
http://www.wolframalpha.com/input/?i=derivative+calculator&f1=%282^x%29%2F%28e^x%29&x=0&y=0&f=Derivative.derivativefunction_%282^x%29%2F%28e^x%29&a=*FVarOpt.1-_**-.***Derivative.derivativevariable---.*--

If you click on 'show steps', the website explain how to solve it.

But I am not the sort to mindlessly copy down answers without understanding them, and I am very stuck on how log(2/e) turns into log(2)-1 when it should be log(2)-log(e). And I don't think this site is wrong, because I have used it for a week now for a lot of derivatives, and it has been on the dot so far.

P.S., sorry for the late reply, was occupied.

agenttiny200 said:

f'(ax) = ax .lna
f(ex) = ex

f(x) = (2x)/(ex)
f(x) = (2/e)x
f'(x) = {(2/e)x} . {log(2/e)}
|
|
|

(THE STEP SHOULDN'T HAVE LOG.IT SHOULD BE LN FUNCTION.)

I hope it's clear now.

ln is to the base e where as log is to the base 10.

Also log(a/b)= log(a)-log(b)

Not log(a)-log(1/b).

This rule can be derived using basic definition of log function.

NOTE :I HAVE EDITED THE QUOTE A LITTLE

Last edited:
okay, maybe the site isn't all that good with log derivatives...I get how it becomes -1 now, thanks!

agenttiny200 said:

agenttiny200 said:
okay, maybe the site isn't all that good with log derivatives...I get how it becomes -1 now, thanks!

The site is fine. Look at the little remark that it says in the first box.

"log(x) is the natural logarithm"

Hey, don't think it that way.
I think Wolfram Alpha is the best site for mathematic calculations.It can do integrations as well

In fact in your answer, look at the first block at the lower right corner.Tou can see that its stated log is natural logarithm :-)

scurty said:
The site is fine. Look at the little remark that it says in the first box.

"log(x) is the natural logarithm"

Same pinch :P
I said the same thing :-)

actually, I just noticed that myself and was about to come back here to correct myself. I swear, this wolframalpha seems to be able to do anything mathematical.

Sorry if the thread is solved but I guess if I write exact calculations,The OP will be better.

$\frac{d}{dx} (\frac{2}{e})^x = \frac{d}{dx} e^{x \ln{\frac{2}{e}}}=\frac{d}{dx} e^ {x (\ln{2}-\ln{e})}=\frac{d}{dx} e^ {x (\ln{2}-1)}=e^{x(\ln{2}-1)} (\ln{2}-1)=(\ln{2}-1) (\frac{2}{e})^x$

## 1. What is the derivative of the natural logarithm function ln(x)?

The derivative of ln(x) is simply 1/x. This can be proven using the definition of a derivative and the properties of logarithms.

## 2. How do you find the derivative of a general logarithm function loga(x)?

The derivative of loga(x) is 1/(xln(a)). This can be derived using the chain rule and the derivative of ln(x).

## 3. What is the derivative of an exponential function ex?

The derivative of ex is simply ex. This can be proven using the definition of a derivative and the properties of the exponential function.

## 4. How do you find the derivative of a general exponential function ax?

The derivative of ax is axln(a). This can be derived using the chain rule and the derivative of ex.

## 5. What is the derivative of a logarithmic function raised to a power, such as (ln(x))2?

The derivative of (ln(x))n is n(ln(x))n-1/x. This can be derived using the chain rule and the derivative of ln(x).

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