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? ::>_<::
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.