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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? ::>_<::

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So, is this a function of x, c and z? of is it just an $f(x)$? – NoChance Nov 5 '12 at 0:50
It is a function of $x$. $c$ and $z$ are constant. – Chang Nov 5 '12 at 1:04
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. – NoChance Nov 5 '12 at 1:17
@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 $x=1$. – Chang Nov 5 '12 at 1:31
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. – NoChance Nov 5 '12 at 2:24
up vote 1 down vote accepted

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.

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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? – Chang Nov 21 '12 at 4:44
@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. – Ross Millikan Nov 25 '12 at 23:13
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? – Chang Nov 26 '12 at 7:45
@Chang: I don't think you will get a closed form, as the functions don't match exactly. When $zx \gg 1$, Alpha 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. – Ross Millikan Nov 26 '12 at 13:56

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