# What are differences between Geometric, Logarithmic and Exponential Growth?

At past I have read in some ecology text that geometrical, logarithmic and exponential growths are not exactly the same thing; and there were various equations for them. (The book is not available to me now, and I forgot its name).

My question is: What is the basic difference of these 3 growth patterns? What would be some real-life analogy to distinguish 3 growth patterns?

Note: This question is Not same with existing decay curve question

Update: I have found a diagram similar to the book in which I saw the concept. Source: https://cmapspublic3.ihmc.us/rid=1R0TPVNFG-113V4JS-1H9C/1R2CJ0126I1VCLY5MI1GHRIimage

Looks like there are already confusion about this distinction between exponential and geometric growth.

Some sources claim a difference such as

There are another source at nature scitable that says

"Exponential growth and geometric growth are similar enough that over longer periods of time, exponential growth can accurately describe changes in populations that reproduce periodically (like bison) as well as those that reproduce more constantly (like humans)"

i.e. it accepts that exponential growth and geometric growth are different at least to some extent.

Some other sources critic this idea such as

this source says this distinction a "Zombie idea"

and according to a quora discussion answer "There is absolutely no difference"

For example, the function 2ˣ tell us that the number 2 can be multiplied “x times” you want. Lets do a simple sequence of the latter function where x goes from 0 to 5. Our sequence looks like this 2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵= 32, … this is exponential growth. Now lets do it using the geometric method that is repeated multiplication, in this case we start with x goes from 0 to 5 and our sequence goes like this: 1, 2, 2•2=4, 2•2•2=8, 2•2•2•2=16, 2•2•2•2•2=32.

The conflicts have made me more confused about the concept of a dfference between Geometric and exponential growth.

• – user379641
Commented Aug 3, 2020 at 2:53

Logarithmic growth and exponential growth is inverse of one another. Difference between exponential growth and geometric growth is that as wikipedia has stated "In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay since the function values form a geometric progression." in https://en.m.wikipedia.org/wiki/Exponential_growth . This means geometric growth is exponential growth but exponential takes real number as it's exponent and geometric just integer implying geometric growth is not continuous as exponential growth is.

• Can I say that the 2 graphs are basically same, and one is just continuous and other is just discrete? If I can find that ecology book (certainly I have wrote its name and publisher some or some place) I will be further able to specify my question.
– user379641
Commented Aug 3, 2020 at 17:32
• The book suggested 2 different graphs as I shown in the attachment.
– user379641
Commented Aug 3, 2020 at 17:33
• @AlwaysConfused The graph in attachment is different since they are different function. Commented Aug 3, 2020 at 17:37

I'll extend a little the scope of your question, presenting some more growth rates. The rates of growth are given by the following formulae: $$\begin{matrix} \text{Logarithmic:} & \log(x)\\ \text{Linear:} & x\\ \text{Polynomial} & x^a\\ \text{Factorial:} & x! \\ \text{Exponential:} & x^x \\ \end{matrix}$$ You can notice that the distinction between exponential and geometric in the texts you found is due to the parameter $$a$$. One property we can stablish is $$\log(n) \ll n \ll a^n \ll n! \ll n^n$$. That means that when $$x$$ goes to infinite and $$a \ll b$$, $$b$$ grows faster than $$a$$ and eventually surpasses it. To check logarithmic, linear vs. polynomial/exponential growth just do the usual line plot. The later will grow much faster than the former. One way of making the distinction between polynomial and exponential is by ploting them in log-scale, the difference of growth becomes more apparent.

A useful instrument to express the growth of a function is big-O notation, an instrument included in Bachmann–Landau notation. We say that a function $$f(x)$$ is $$\mathcal{O}(g(x))$$ when the following conditions is satisfied.

$$\exists M\in \mathbb{R},\; x_0\in \mathbb{R} \; \text{s.t.} \; \forall x>x_0, \left\lvert f(x) \right\rvert < Mg(x)$$

That is a function that is $$\mathcal{O}(g(x))$$ is bounded by another function $$g(x)$$ times a real constant. This gives some sort of growth boundaries for a function. It increases or decreases as much or less than $$g(x)$$.

• So if something increased via a power law, say $t^{3/2}$ would one call it "power law growth"?
– uhoh
Commented Aug 20, 2020 at 8:49
• @uhoh I guess you can use the expression "power law growth", however I usually see the term power law also assigned to distributions like the Pareto distribution. I would say "polynomial growth" as it is more general/standard. Notice that the expression you provided is equivalent to poly case when $a=\frac{3}{2}$. Commented Aug 20, 2020 at 22:06
• @uhoh I mean being really rigorous we should call it another name, howeve I've seem people extend to rational exponents and keep only the non-negative bit like here: people.sc.fsu.edu/~pbeerli/classes/ISC4221/ISC4221/Class_Slides/… One thing that I didn't mention in my answer is big O notation. You can assign to a function a sort of a growth rate through upper bounds, like this $\mathcal{O}(x^a)$. Commented Aug 22, 2020 at 17:04
• I'll add some comments about big-O notation to my answer. Commented Aug 22, 2020 at 17:07
• I'm always in favor of anything that leads to a big O.
– uhoh
Commented Aug 22, 2020 at 22:33