How Natural Is The Natural Logarithm?

I want to show in this post that the natural logarithm is not natural – it is not a characteristic of objective nature.

The natural logarithm pops up everywhere in science: biology, sociology, economics… and the list goes on. Every student of physics knows that many of the calculations in physics (for example in electrodynamics and quantum mechanics) would be undoable if it weren’t for the natural logarithm. The universal applicability of the natural logarithm suggests that it is something that exists in the world in which we live; that it is a characteristic of the physical world. We will see, however, that the naturalness of the natural logarithm (having base $e$) is a characteristic of the definition of derivative – and of any logarithm to which this definition can be applied.

Wherever there’s derivation, there’s $\bold{e}$

The natural logarithm is a logarithm with as its base the mathematical constant $e$. $e$ is a number such that the function $e^t$ for every $t$ has a value that is equal to its rate of growth $[e^t=\frac{de^t}{dt}]$. The existence of a number with that characteristic follows logically from our definition of derivative.

The derivative of a function $f(t)$ is defined as follows:

$\frac{df(t)}{dt}=\lim_{h \to0}\frac{f(t+h)-f(t)}{h}$.

In mathematics all exponential functions are of the form $f(t)=ca^t$  Therefore, if $f(t)$ is an exponential function then the following holds:

$\frac{df(t)}{dt}=\lim_{h\to0}\frac{ca^{t+h}-ca^t}{h}$

We can move $ca^t$ outside the limit so that the limit no longer depends on the variable $t$ (and it is therefore a constant in $f(t)$):

$\frac{dca^t}{dt}=\lim_{h\to0}\frac{ca^{t+h}-ca^t}{h}=ca^t[\lim_{h\to0}\frac{a^{h}-1}{h}]$.

Let’s take a closer look at the latter limit. In it there are the two functions $f_1(h)=a^h-1$ and $f_2(h)=h$. We may calculate the value of the limit with the help of l’Hôpital’s rule:  by determining the derivatives of $f_1$ and $f_2$ and computing $\frac{f_1'}{f_2'}(0)$. Determining $f_2'$ is easy; $f_2'=\frac{dh}{dh}=1$. The value of $f_1'$ is less straightforward, because it depends on the value of $a$. If $a$ in $f_1$ has the value $e$ then $\frac{f_1'}{f_2'}(0)=\frac{e^h}{1}(0)=1$ (in the figure you can see that for $h=0$ $f_1$ and $f_2$ have both the same value and the same slope; $f_1'$ and $f_2'$ also have the same value). Returning with our findings about this limit to the equation for $\frac{da^t}{dt}$ we see that the only exponential function for which $\frac{df(t)}{dt}=f(t)$ is the exponential with base $e$; $\frac{dca^t}{dt}=ca^t$ only if $a=e$.

Every exponential can be expressed as a natural exponential

Consider again the general form of the exponential function, $f(x)=ca^x$. There are two ways in which we can transform any function of the form $ca^x$ into a natural exponential function.

1. Firstly, we can choose a suitable variable transformation ($t \rightarrow x$, where $x=\log_a e^t$) so that we may write $f(x)=ca^x$ as $f(t)=ce^t$.
2. Alternatively, if we don’t want to adjust the variable, we may rewrite $ca^x$ as $ce^{c_1x}$ (for $a=e^{c_1}$) and then let $e^{c_1}$ merge with $c$ into some other constant $c_2$ so that we end up with $f(x)=c_2e^x$

Suppose for example that a biologist, let’s call her Fleur, studies a colony of bacteria in a Petri dish. Fleur finds that the number of bacteria in the Petri dish grows exponentially with a linear increase in time. Say she counts the time, $t$, in seconds and formulates the function $n(t)=ca^t$ to describe the number of bacteria. If $a\neq e$ then n(t) is cumbersome to work with mathematically. If Fleur could rewrite $n(t)$ in terms of a natural logarithm then any differential equation with $n(t)$ in it would become much easier to solve.

Suppose Fleur indeed wants to rewrite $n(t)$ in terms of a natural logarithm. She has measured the number of bacteria at $t=0$ and found it to be $n(0)=c$. It would be a bad idea for Fleur to use the second method (2) of naturalising $n(t)$ because that would change $n(0)$ and then $n(t)$ would conflict with her initial measument. Fleur, being the clever biologist that she is, realises that and therefore opts for method (1). Suppose that she sees that $n(t)$ doubles every 2.5 seconds. Then she might try substituting the variable $t$ for a variable $x$ for which each unit-step is not 1 second but 2.5 seconds ($x=\frac{1}{2.5}t$). The function $n(t)$ can now be rewritten into a function $n(x)$ which has a natural exponential form: $n(x)=ce^x$. This will make Fleur’s life much easier.

Now we may ask the question whether the natural exponential growth-rate is a characteristic of the system that is studied or a characteristic of the way we describe that system. The facts that $e$ is a characteristic of the definition of derivative and that any exponential function can be written as a natural exponential might suggest that $e$ is a characteristic of our number system and not of nature. Such a conclusion, however, disregards a distinguishing feature of logarithmic functions.

But not every phenomenon can be described by an exponential

The distinguishing feature of logarithmic functions is that their rate of growth at a certain moment scales with the value of the function at that moment. A natural logarithmic function is a special case of this in that the scaling is very simple: the rate of growth is at every moment exactly equal to the value of the function at that moment.

Any dataset that can be described in terms of exponentials can also be described in terms of natural exponentials (any exponential function can be transformed such that its base is $e$), but we have stumbled upon an objective fact about nature (as it comes to us in our observations) whenever we discover that some phenomenon allows a description in terms of exponentials at all.

The fact that we can describe nature in terms of natural logarithms is a consequence of the way we describe nature, but the fact that we can use logarithms at all tells us something about nature’s dynamics.

About fbenedictus

Philosopher of physics at Amsterdam University College and Utrecht University, managing editor for Foundations of Physics and international paraclimbing athlete
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2 Responses to How Natural Is The Natural Logarithm?

1. F.A. Muller says:

Very nice.

2. Good old discouvery vs. invention of math discussion! Of course, a lot more can be said about $e^x$. Maybe, the most interesting in this discussion would be its use in describing wave-functions. Or would you argue that the sine and cosine functions are also, just a consequence of our description of nature? You could go on nihilating the meaning of these arguably purely mathematical pieces of knowledge, and ultimately arrive at a meaningless solipsistic description of your own little universe. $e^{i\pi} = -1$, Euler salutes you!