What are differences between Geometric, Logarithmic and Exponential Growth?

Question

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

Thanks in advance.

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.

Answer 1

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.

Answer 2

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 $g(x)$.