# Making logarithmic function go higher

I am looking at logarithmic functions, and, lets say, log2 (x+3) is having a bit of a growth rate between 0-10 values of x, but then it has almost no growth rate.

I need some function that looks similar to a linear function, but is slightly bent like a logarithm function is.

How do I modify my logarith function?

Edit: I made something like this: http://www.wolframalpha.com/input/?i=graph+log_100%28x%5E128%29++from+x%3D2+to+x%3D4000 and I have almost the results I want, but I would like this function to not go as extreme high in first 0-500 values of x axis.

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You are aware that $\log x^{128} = 128\log x$, so the function you've linked is just some scaled version of $\log x$? –  martini Sep 26 '13 at 14:51
@martini: Wasn't aware, of that, thanks. ;) But I need to modify it somehow so it will get "wider" at the beginning, and I don't know how to do that. –  ojek Sep 26 '13 at 14:57

Modification: $$a-a^{1-\frac{x}{b}}.$$
This starts at the origin, and quickly converges to $a$. After a point there is relatively speaking little growth. Try making $a=10$, and $b=5$, and then setting your aspect ratio to 1 in a better grapher (for example ojek graph allows manipulate constructions).