# Can I flip the exponential growth function to get a logarithmic growth function

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:

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?

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.

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:

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.

• What do you mean by log-growth? Does it mean that the function itself is a log-function or the derivative of the function? May 21 at 16:20
• 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? May 21 at 16:22
• @callculus I mean that the shape of the curve is logarithmic. See my edit. Does my post make more sense now? May 21 at 16:52
• @Vepir that's a great link! I've just bookmarked it. I guess I'd be looking at the one titled 'Logarithmic Model. May 21 at 16:53
• In fact I think @Vepir's link answers my question May 21 at 16:57

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 $$1.386294$$