Results 1 to 8 of 8
Like Tree5Thanks
  • 1 Post By Melody2
  • 2 Post By ibdutt
  • 1 Post By Melody2
  • 1 Post By Melody2

Math Help - Natural Log of X^x.

  1. #1
    Member
    Joined
    May 2011
    Posts
    133
    Thanks
    2

    Natural Log of X^x.

    f=x^x
    ln(f)=x*ln(x)

    I am aware of the proof that:

    \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.

    Thanks in advance...
    Last edited by sepoto; November 17th 2013 at 05:59 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    192
    Thanks
    39

    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.
    Thanks from sepoto
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2011
    Posts
    133
    Thanks
    2

    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.
    Last edited by sepoto; November 17th 2013 at 07:17 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jul 2012
    From
    INDIA
    Posts
    829
    Thanks
    209

    Re: Natural Log of X^x.

    Natural Log of X^x.-18-nov13-2.png
    Thanks from sepoto and topsquark
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    May 2011
    Posts
    133
    Thanks
    2

    Re: Natural Log of X^x.

    I recognize:
    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:

    1*ln(x)+

    Thank you all for your replies.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    192
    Thanks
    39

    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

    which might be helpful.

    I'm sorry I've been slow to answer I wanted to use Latex but I couldn't get it to work properly.
    Thanks from sepoto
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    May 2011
    Posts
    133
    Thanks
    2

    Re: Natural Log of X^x.

    It appear that the derivative of x*ln(x) is ln(x)+1. Thanks for posting...
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    192
    Thanks
    39

    Re: Natural Log of X^x.

    \begin{align*}\dfrac{d}{dx}\;(x\times lnx) &= 1\times lnx + x\times \dfrac{1}{x}\\&=lnx+1\end{align*}

    I used the product rule.
    Thanks from sepoto
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. natural log help
    Posted in the Algebra Forum
    Replies: 2
    Last Post: August 8th 2012, 02:54 AM
  2. Fun with natural log
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: September 13th 2010, 09:05 AM
  3. simplify natural log times natural log
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: March 3rd 2010, 09:31 PM
  4. natural log
    Posted in the Pre-Calculus Forum
    Replies: 7
    Last Post: September 1st 2009, 10:16 AM
  5. HELP W/ Natural Log
    Posted in the Algebra Forum
    Replies: 1
    Last Post: June 9th 2008, 10:22 PM

Search Tags


/mathhelpforum @mathhelpforum