0
$\begingroup$

Recently I am thinking about this question:

Assume $x$ is real, $x\geq0$, $c$ is a positive constant number and $z$ is also a real constant between $3.5$ and $4$. Now there is a function: $$ f(x)=\frac{x}{c+\frac{1}{1-\left(1+\frac{1}{zx}\right)^{-z}}}. $$ I want to find whether there is an approximation for $f(x)$ when $x$ is between $\left[0.1, 10\right]$ by logarithm function, because I draw the figure and it looks like it... (So, I just guess...).

The reason I want to find an approximation is because the expression of $f(x)$ is complicated. And from the figure, it is really like a logarithm function near $x=1$.

Could you help me? I hope to discuss with you. Thank you in advance.

Can anyone help? ::>_<::

  • 0
    So, is this a function of x, c and z? of is it just an $f(x)$?2012-11-05
  • 0
    It is a function of $x$. $c$ and $z$ are constant.2012-11-05
  • 0
    If $x, z$ are constants, then what you have is $f(x)=x/K$ where $K$ is constant. This is a straight line and not a log function.2012-11-05
  • 0
    @Emmad Kareem, thank you for your reply. Could you explain in more detail? I know that when $x\rightarrow0$ or $x\rightarrow\infty$, $f(x)$ is linear with $x$. But I don't understand how you get the linearity near $x=1$.2012-11-05
  • 0
    Please ignore my comment, I did not see that you have (1/zx). I was viewing this on a tiny pad. This is not linear at all.2012-11-05
  • 0
    @Emmad Kareem, thank you for your reply. So, do you have any clues on this?2012-11-06
  • 0
    I am not able to find an approximation for this form.2012-11-06

1 Answers 1

1

When you say "logarithm function" you need to specify what is in that class. Is it $g(x)=\log_y x$? Or something with more variables, more "knobs to turn", like a sum of two logs, or $\log_y (x-a)$. You can certainly use a function minimizer, multidimensional if necessary, to minimize $\int_{0.1}^{10}(g(x)-f(x))^2 dx$, then plot $f$ and $g$ to see if the agreement is to your liking. Any numerical analysis text can give you pointers. I like chapter 10 of Numerical Recipes, the obsolete versions are free on line.

  • 0
    Well, thank you for your reply. I used Matlab and find that $f(x)$ can be approximated by $g(x)=c_1\log_{10}(x)+c_2$. Now the difficulty is to find the $c_1$ and $c_2$ by using $c$ and $z$. I will check the chapter you mentioned. Thank you. And do you have any further suggestions?2012-11-21
  • 0
    @Chang: this is a 2D linear minimization problem now. The fact that $\log{10} x$ is nonlinear does not matter, what matters is that your approximation is linear in the parameters-$c_1$ and $c_2$. I'm sure Matlab has one.2012-11-25
  • 0
    Thank you for your answer. I know using Matlab can help me to find the numerical result. But I hope to find the closed form solution of $c_1$ and $c_2$ w.r.t. $c$ and $z$. I am not sure whether Matlab could do that?2012-11-26
  • 0
    @Chang: I don't think you will get a closed form, as the functions don't match exactly. When $zx \gg 1$, [Alpha](http://www.wolframalpha.com/input/?i=Expand+1-%281%2B1%2F%28zx%29%29^%28-z%29+in+x) gives $1-(1+\frac 1{xz})^{-z} \approx \frac 1x-\frac {z+1}{2 x^2 z}$. But unless you are using a calculator, this expression is not bad-a few lines of code.2012-11-26