# Natural Log of X^x.

• Nov 17th 2013, 05:39 PM
sepoto
Natural Log of X^x.
$\displaystyle f=x^x$
$\displaystyle ln(f)=x*ln(x)$

I am aware of the proof that:

$\displaystyle \frac{d}{dx}a^x=ln(a)*a^x$

but ln(f)=x*ln(x) seems to be following a different format since it would have to be the quantity x^x*ln(x) however my lecture notes seem to be saying that ln(f)=x*ln(x)?

Also with two x's it gets confusing on which one is the exponent and which is the base.

• Nov 17th 2013, 06:58 PM
Melody2
Re: Natural Log of X^x.
I haven't thought this through very well yet but the really big difference is that 'a' is a constant.
The proof that you know only deals with 'a' as a constant.
• Nov 17th 2013, 07:09 PM
sepoto
Re: Natural Log of X^x.
So as I see it ln was taken to both sides. So then ln of x^x is equal somehow to x ln(x) it says in the lecture notes. X^x transformed to ln which is base e. I think the key to this is to ask. What is the natural log of X^x? Or in my case I'm wondering what are the rules to taking the natural log of the exponent x^x? I ask because if it was shown to me what is the natural log of that exponent then I don't remember where.

I dug this out which I think has the answer:
Natural logarithm rules - ln(x) rules

It looks a lot like the logarithm power rule.
• Nov 17th 2013, 07:16 PM
ibdutt
Re: Natural Log of X^x.
• Nov 17th 2013, 07:32 PM
sepoto
Re: Natural Log of X^x.
I recognize:
$\displaystyle x*\frac{1}{x}$
as the derivative of x ln(x). Although it looks like the chain rule is working here I have to admit I am still trying to understand where the part below came from:

$\displaystyle 1*ln(x)+$

Thank you all for your replies.
• Nov 17th 2013, 07:38 PM
Melody2
Re: Natural Log of X^x.
So then ln of x^x is equal somehow to x ln(x) it says in the lecture notes.
Yes this is absolutely true (it doesn't have to be base e it will work the same with any base)

When you take the log of both sides of an equation to solve it you can use ANY base. It doesn't matter which one.
You would normally use base e because it is friendly for differentiation or because you can estimate the answer with your calculator.

There is a list of log identities in wikipedia
List of logarithmic identities - Wikipedia, the free encyclopedia

and also an interactive learning site in mathsisfun
Introduction to Logarithms

It appear that the derivative of $\displaystyle x*ln(x)$ is $\displaystyle ln(x)+1$. Thanks for posting...
\displaystyle \begin{align*}\dfrac{d}{dx}\;(x\times lnx) &= 1\times lnx + x\times \dfrac{1}{x}\\&=lnx+1\end{align*}