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High school textbook exponential growth:

$$A(t)=A^{kt}$$

Where $A$ is the initial amount, $t$ is the number of time periods and $k$ is the growth rate. With this formula I can determine the function for a given variable using two known points. Then, once I have the function I can estimate the amount after $t$ time periods.

I wondered if there's an equivalent logarithmic function? I'm working with a variable that appears to follow log growth and I would like to know if there's a function I can use, similar to exponential growth, where I can plug in an initial amount $A$ and estimate the future amount $A(t)$ after t time periods. I would like to know if I can estimate $k$ in the same way as with a exponential model?

I was thinking I could try to flip a exponential line to get it logarithmic.

E.g. just playing around with the exponential function on Desmos, I can swap signs and add fixed amounts to get a line resembling a looking closer to a logarithmic function, e.g. $y=-10^{-\left(0.5\cdot t-1\right)}+10$

Looks like this: enter image description here

Is it possible to wrangle the exponential growth formula to make it work for logarithmic growth where I can estimate future values of $Y$ after $t$ time periods?

[Edit following comments]

Here's a visual of the variable I'm working with. It's cumulative revenue for an app. Data are disguised and scaled. It appears, visually, like a logarithmic curve.

enter image description here

Here's the same chart with the natural log of the x variable. Not a perfectly straight line but for the sake of this post let's pretend that it is: enter image description here

It's logarithmic growth in that is starts of steep before leveling off. As opposed to exponential growth which starts of slowly before sharply increasing.

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  • $\begingroup$ What do you mean by log-growth? Does it mean that the function itself is a log-function or the derivative of the function? $\endgroup$ May 21 at 16:20
  • $\begingroup$ I do not understand what you are trying to accomplish? What thing of the things listed at Exponential and Logarithmic Models are you trying to model? Do you mean Logisitic or Logarithmic? $\endgroup$
    – Vepir
    May 21 at 16:22
  • $\begingroup$ @callculus I mean that the shape of the curve is logarithmic. See my edit. Does my post make more sense now? $\endgroup$
    – Doug Fir
    May 21 at 16:52
  • $\begingroup$ @Vepir that's a great link! I've just bookmarked it. I guess I'd be looking at the one titled 'Logarithmic Model. $\endgroup$
    – Doug Fir
    May 21 at 16:53
  • $\begingroup$ In fact I think @Vepir's link answers my question $\endgroup$
    – Doug Fir
    May 21 at 16:57
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If you want to use $y=a+b\cdot\ln(x)$ you can apply a linear regression to estimate $a$ and $b$. But first you have to transform the x values.

$$\ln(x)=z$$

Then the regression line becomes $y=a+b\cdot z$

Example: If the $x$ values are $1,2,3$ and $4$ then the $z$-values are

$0, 0.693147, 1.098612$ and $1.386294$

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