What is x in terms of p?












0












$begingroup$


In this equation




$$frac x{ln(x)} = p$$




How can I have $x$ as a function of $p$?






logarithms inverse-function







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    I don't think the function $f(x)=frac{x}{ln x}$ has an elementary inverse. That is, the function is invertible, but there is no way to "nicely" write the inverse of it.
    $endgroup$
    – 5xum
    Jan 14 at 13:53






  • 2




    $begingroup$
    Assuming $p$ is an integer, $x$ is almost the $p$th prime, if that helps.
    $endgroup$
    – Arthur
    Jan 14 at 13:55


















0












$begingroup$


In this equation




$$frac x{ln(x)} = p$$




How can I have $x$ as a function of $p$?






logarithms inverse-function







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    I don't think the function $f(x)=frac{x}{ln x}$ has an elementary inverse. That is, the function is invertible, but there is no way to "nicely" write the inverse of it.
    $endgroup$
    – 5xum
    Jan 14 at 13:53






  • 2




    $begingroup$
    Assuming $p$ is an integer, $x$ is almost the $p$th prime, if that helps.
    $endgroup$
    – Arthur
    Jan 14 at 13:55
















0












0








0





$begingroup$


In this equation




$$frac x{ln(x)} = p$$




How can I have $x$ as a function of $p$?






logarithms inverse-function







share|cite|improve this question











$endgroup$




In this equation




$$frac x{ln(x)} = p$$




How can I have $x$ as a function of $p$?






logarithms inverse-function




logarithms inverse-function






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 14 at 13:57









mrtaurho

5,44541237




5,44541237










asked Jan 14 at 13:51









Matin N.Matin N.

51




51








  • 1




    $begingroup$
    I don't think the function $f(x)=frac{x}{ln x}$ has an elementary inverse. That is, the function is invertible, but there is no way to "nicely" write the inverse of it.
    $endgroup$
    – 5xum
    Jan 14 at 13:53






  • 2




    $begingroup$
    Assuming $p$ is an integer, $x$ is almost the $p$th prime, if that helps.
    $endgroup$
    – Arthur
    Jan 14 at 13:55
















  • 1




    $begingroup$
    I don't think the function $f(x)=frac{x}{ln x}$ has an elementary inverse. That is, the function is invertible, but there is no way to "nicely" write the inverse of it.
    $endgroup$
    – 5xum
    Jan 14 at 13:53






  • 2




    $begingroup$
    Assuming $p$ is an integer, $x$ is almost the $p$th prime, if that helps.
    $endgroup$
    – Arthur
    Jan 14 at 13:55










1




1




$begingroup$
I don't think the function $f(x)=frac{x}{ln x}$ has an elementary inverse. That is, the function is invertible, but there is no way to "nicely" write the inverse of it.
$endgroup$
– 5xum
Jan 14 at 13:53




$begingroup$
I don't think the function $f(x)=frac{x}{ln x}$ has an elementary inverse. That is, the function is invertible, but there is no way to "nicely" write the inverse of it.
$endgroup$
– 5xum
Jan 14 at 13:53




2




2




$begingroup$
Assuming $p$ is an integer, $x$ is almost the $p$th prime, if that helps.
$endgroup$
– Arthur
Jan 14 at 13:55






$begingroup$
Assuming $p$ is an integer, $x$ is almost the $p$th prime, if that helps.
$endgroup$
– Arthur
Jan 14 at 13:55












1 Answer
1






active

oldest

votes


















3












$begingroup$

This is a typical example of a transcendental equation with a solution that cannot be expressed with most widely known analytical functions (trigonometric, exponential and logarithmic).



Rule of thumb: if $x$ is found both inside and outside log/exp/sin, the equation cannot be inverted in terms of these functions.



However, the equation is common enough to have a so called "special function" which can be used to invert it: Lambert's function $x=W(y)$, which solves equation $xe^{x}=y$.



Of course, as Lambert's function doesn't usually have a button on a calculator, that does not help much, but does mean people have studied this function and have efficient means of calculating its value.



You just have to transform it into the form to apply the rule for Lambert's function;



$$ln x = frac{x}{p}$$
$$x=e^{x/p}$$
$$xe^{-x/p}=1$$
new variable $u=-x/p$
$$-pu e^{u}=1$$
$$ue^u=-1/p$$
$$u=W(-1/p)$$
$$x=-pu=-p W(-1/p)$$



Of course you have to be careful because for negative arguments, $W$ can have two branches (equation has two solutions), or no real solution at all, if the value is too negative (below $-1/e$).






share|cite|improve this answer











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    1 Answer
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    1 Answer
    1






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    active

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    3












    $begingroup$

    This is a typical example of a transcendental equation with a solution that cannot be expressed with most widely known analytical functions (trigonometric, exponential and logarithmic).



    Rule of thumb: if $x$ is found both inside and outside log/exp/sin, the equation cannot be inverted in terms of these functions.



    However, the equation is common enough to have a so called "special function" which can be used to invert it: Lambert's function $x=W(y)$, which solves equation $xe^{x}=y$.



    Of course, as Lambert's function doesn't usually have a button on a calculator, that does not help much, but does mean people have studied this function and have efficient means of calculating its value.



    You just have to transform it into the form to apply the rule for Lambert's function;



    $$ln x = frac{x}{p}$$
    $$x=e^{x/p}$$
    $$xe^{-x/p}=1$$
    new variable $u=-x/p$
    $$-pu e^{u}=1$$
    $$ue^u=-1/p$$
    $$u=W(-1/p)$$
    $$x=-pu=-p W(-1/p)$$



    Of course you have to be careful because for negative arguments, $W$ can have two branches (equation has two solutions), or no real solution at all, if the value is too negative (below $-1/e$).






    share|cite|improve this answer











    $endgroup$


















      3












      $begingroup$

      This is a typical example of a transcendental equation with a solution that cannot be expressed with most widely known analytical functions (trigonometric, exponential and logarithmic).



      Rule of thumb: if $x$ is found both inside and outside log/exp/sin, the equation cannot be inverted in terms of these functions.



      However, the equation is common enough to have a so called "special function" which can be used to invert it: Lambert's function $x=W(y)$, which solves equation $xe^{x}=y$.



      Of course, as Lambert's function doesn't usually have a button on a calculator, that does not help much, but does mean people have studied this function and have efficient means of calculating its value.



      You just have to transform it into the form to apply the rule for Lambert's function;



      $$ln x = frac{x}{p}$$
      $$x=e^{x/p}$$
      $$xe^{-x/p}=1$$
      new variable $u=-x/p$
      $$-pu e^{u}=1$$
      $$ue^u=-1/p$$
      $$u=W(-1/p)$$
      $$x=-pu=-p W(-1/p)$$



      Of course you have to be careful because for negative arguments, $W$ can have two branches (equation has two solutions), or no real solution at all, if the value is too negative (below $-1/e$).






      share|cite|improve this answer











      $endgroup$
















        3












        3








        3





        $begingroup$

        This is a typical example of a transcendental equation with a solution that cannot be expressed with most widely known analytical functions (trigonometric, exponential and logarithmic).



        Rule of thumb: if $x$ is found both inside and outside log/exp/sin, the equation cannot be inverted in terms of these functions.



        However, the equation is common enough to have a so called "special function" which can be used to invert it: Lambert's function $x=W(y)$, which solves equation $xe^{x}=y$.



        Of course, as Lambert's function doesn't usually have a button on a calculator, that does not help much, but does mean people have studied this function and have efficient means of calculating its value.



        You just have to transform it into the form to apply the rule for Lambert's function;



        $$ln x = frac{x}{p}$$
        $$x=e^{x/p}$$
        $$xe^{-x/p}=1$$
        new variable $u=-x/p$
        $$-pu e^{u}=1$$
        $$ue^u=-1/p$$
        $$u=W(-1/p)$$
        $$x=-pu=-p W(-1/p)$$



        Of course you have to be careful because for negative arguments, $W$ can have two branches (equation has two solutions), or no real solution at all, if the value is too negative (below $-1/e$).






        share|cite|improve this answer











        $endgroup$



        This is a typical example of a transcendental equation with a solution that cannot be expressed with most widely known analytical functions (trigonometric, exponential and logarithmic).



        Rule of thumb: if $x$ is found both inside and outside log/exp/sin, the equation cannot be inverted in terms of these functions.



        However, the equation is common enough to have a so called "special function" which can be used to invert it: Lambert's function $x=W(y)$, which solves equation $xe^{x}=y$.



        Of course, as Lambert's function doesn't usually have a button on a calculator, that does not help much, but does mean people have studied this function and have efficient means of calculating its value.



        You just have to transform it into the form to apply the rule for Lambert's function;



        $$ln x = frac{x}{p}$$
        $$x=e^{x/p}$$
        $$xe^{-x/p}=1$$
        new variable $u=-x/p$
        $$-pu e^{u}=1$$
        $$ue^u=-1/p$$
        $$u=W(-1/p)$$
        $$x=-pu=-p W(-1/p)$$



        Of course you have to be careful because for negative arguments, $W$ can have two branches (equation has two solutions), or no real solution at all, if the value is too negative (below $-1/e$).







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 14 at 14:52

























        answered Jan 14 at 13:56









        orionorion

        13.6k11837




        13.6k11837






























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