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June 14th, 2018, 08:03 PM  #1 
Newbie Joined: Dec 2016 From: California Posts: 9 Thanks: 0  Function for saturating logarithmic growth
Hi everyone, After searching for a bit, I'm wondering: is there a class of functions which exhibit logarithmic growth and then saturate? An integrable form would be ideal. It seems like such a simple scenario, but I'm coming up short. Thanks! 
June 15th, 2018, 02:07 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,626 Thanks: 622 
arctan(x) for x > 0 might work. At x=0 it is linear, but then "saturates" to $\frac{\pi}{2}$.

June 15th, 2018, 08:34 PM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 2,165 Thanks: 1139 
The function can't be strictly logarithmic growth if it is to saturate. Can you separate the domain into an area where you want true logarithmic growth and then one that can tolerate the saturation? 
June 18th, 2018, 09:36 AM  #4 
Newbie Joined: Dec 2016 From: California Posts: 9 Thanks: 0 
Thanks for the responses! mathmanThis is an interesting suggestion. arctan(x) does exhibit the sort of saturation that I'm interested in describing, but below saturation the growth is linear on a loglog plot, whereas I'm looking for a function which is linear on a linear y / log x plot (i.e., logarithmic growth). romsekYes, there are two domains, though they should grade into each other (not a break in slope, but a gradual transition). I may have cobbled together a workable solution: y = ln(1  exp[x]) 
June 21st, 2018, 04:27 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
Sounds to me like you are talking about the "logistic function": Logistic Functions 
June 22nd, 2018, 10:41 AM  #6 
Newbie Joined: Dec 2016 From: California Posts: 9 Thanks: 0 
I don't think so, only because logistic functions begin to grow in an exponential manner (increasing growth rate with time) before they begin to saturate. What I'm describing is function which is always decreasing in growth rate but which then saturates completely (unlike traditional logarithimic growth, which increases forever, though ever more slowly). That said, you could be rightthere could be a subcategory of logistic fxn which behaves like you describe that I'm unfamiliar with. 
June 22nd, 2018, 11:06 AM  #7 
Senior Member Joined: Sep 2015 From: USA Posts: 2,165 Thanks: 1139 
I'd pick some point $x_b$ you're happy with being the boundary of the domains. Then $f(x) = \begin{cases} c_1 \log(a x) &x <= x_b \\ \\ c_2 \left(1e^{b x}\right) &x_b < x \end{cases}$ where $c_1 \log(a x_b) = c_2 \left(1e^{b x_b}\right)$ You'll need to tweak $a,~b,~c_1$ to get exactly the behavior you want. You may also want to provide a shift for $x$ i.e. make $f(x) \to f(xx_0)$ where you choose $x_0$ to keep things away from $x<1$ I think you'll find this does the trick. 
June 22nd, 2018, 02:34 PM  #8 
Newbie Joined: Dec 2016 From: California Posts: 9 Thanks: 0 
Thanks very much, romsek. This is a really clean solution!


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