Where we learn that we each have
an internal power function ready to bust out.
Let’s begin with this excellent definition of what a logarithm is:
I find that the clearest way to explain what a logarithm is is to first say what its relationship is to an exponent, because the word exponent (noun) and the word exponential (adjective) are much more familiarly used in our daily lexicon. An exponent is that tiny hanger-on up in the top corner next to a base number, dictating how many times that base number should be multiplied by itself:
34 = 81
Exponents do their work pretty quickly – you know, exponentially 😉- and I feel like their drama is so In Your FACE just because their operation feels and actually gives a visual impression that is practically instantaneous. You know the type. It’s like, Mr. 4 Exponent showed up, located where 3 was in the room, and then he jumped high and stood on that table and attached himself to 3, and then – where there was just 3, whooooosh! Boom! 3 is now 81!!
The inverse function of an exponent is a logarithm, so it essentially performs the opposite function of the exponent’s operation. That logarithm is the undo’r you invite when things are overwhelming, when you’re starting with a big number and you just need to compress and make things manageable (again) so that you can pause, take a breath, and get a sense of gaining (back) some control of the situation at hand.
That same example above but now in logarithmic form looks like this:
Log3 81 = 4
So, if your goal is to control that whoosh! Boom! situation at hand that Mr. 4 Exponent has just wrought upon everyone, then you’d invite Ms. 3 Log, who’s going to calmly come in and take a seat next to where 81 is standing (who is taking up a lot of space in that room) – and just like that aaaahhh, 81 is gone and 4 is now like all the other base numbers in the room; they’re not standing on a table nor are they sitting on a chair, and 3 is back to how they were as if they never met that Mr 4 Exponent guy.
You see – where the exponent is blowin’ things up, the logarithm is like, let’s do deep breath exercises, namaste.
The word logarithm was coined by the Scottish mathematician John Napier, from the two ancient Greek terms: logos meaning proportion, and arithmos meaning number.
A few years back, mathematician Steven Strogatz wrote a series of math essays for the New York Times, and in his essay titled “Power Tools”, he gave an explanation for the very existence of logarithms.
We need them because it’s always useful to have tools that can undo one another. Just as every office worker needs both a stapler and a staple remover, every mathematician needs exponential functions and logarithms.“Power Tools” by Steven Strogatz
NYT Opinionator column
When I read that, I realized that sometimes I feel like a logarithm.
It’s definitely how I feel when I am starting an organizational evaluation of a company’s framework. In my initial steps, as I collect representational data from across the breadth of an entire company, I bring it down to a manageable state for clear universal understanding; I compress the information. That being said, I don’t omit any information because I know that it’s imperative for the final analysis to not minimize-the-existence-of the parts that make up the greater whole and all (the people) within it.
Another thing about logarithms is that the general public isn’t very familiar with them, certainly not as familiar as they are with an exponent. You know you’ve seen a Superscript used in an equation, perhaps when you were reading about the spread of an infectious disease, but how often have you seen a Subscript in a news article? We usually have an innate preference towards the visually familiar, rather than spending the time it might take to comprehend something new. We might even be dismissive about the thing we aren’t familiar with because why take the energy to be curious about that new thing if the familiar thing will serve the needs well enough? We make conclusions that it’s probably of lesser value.
And that’s another reason why I sometimes feel like a logarithm;
it’s when I realize I’ve been underestimated by others.
Like the Superscript and Subscript example, sticking with what you’re familiar with is easy, especially if your reference points for the unfamiliar can’t be updated without some substantial time & effort. I’m constantly surprised how some people will say they are curious and want to learn about a human identity that they’re unfamiliar with, yet – in reality – the patience and time needed to unlearn a belief is devalued. So, a bias (implicit or explicit) is developed, usually from some mass impersonal reference point, for that unfamiliar thing – and definitive conclusions are made. I have been immediately underestimated because I look younger than I am (which, to some folks, then equates to being inexperienced), or because I have visible tattoos (and you know what that means 🙄), or simply because I am a woman ( 🙄etc ).
Or, sometimes I feel like a logarithm because of its power of perception.
Inline with those assumptions that are based on biases, perception is a close relative. In the 2014 edition of “The Best Writing on Mathematics” book, an annual publication of math essays that is edited by Mircea Pitici, right from the beginning of the essay “Why Do We Perceive Logarithmically?” written by Lav R. Varshney and John Z. Sun, the authors’ setup is spot-on:
Why do small children place 3 halfway between 1 & 10? Why do two lightbulbs not seem twice as bright as one? Why do we perceive logarithmically?from the essay, “Why Do We Perceive Logarithmically?”
by Lav R. Varshney and John Z. Sun
The Best Writing on Mathematics, 2014
What struck me in their essay is when they explain a term that I had never heard before – psychophysical – which is a branch of psychology about the direct relationship between our internal / mental processes (the psychic) and our external / stimulation processes (the physical), and how the default conclusion we tend to end up at is logarithmically based. Further explanation from the essay:
“But in fact perceptual systems of all kinds display a non-linear relationship between external stimulus and internal representation…If we double the force on your hand, it will feel like less than double the pressure. If we double the salinity of water, the taste will not be twice as salty.”from the essay, “Why Do We Perceive Logarithmically?”
by Lav R. Varshney and John Z. Sun
The Best Writing on Mathematics, 2014
The reason we (all) tend to do this sometimes is because “we do not notice absolute changes in stimuli; we notice relative changes.” What’s happening is that we perceive what we see and hear logarithmically.
This personal logarithmic perception realization may be more about me being a logarithmic interpreter. When I worked in 3D animation, I’d often be the go-between an artist and a client. A client might ask for the artist to ‘double the yellow color.’ And the artist would go ahead and double the intensity of that color, and we’d send it back to the client for review, and then the client would invariably say something like, “you didn’t double it, I barely see the difference.”
And that’s how those of us in the studio learned that when there’s a numerically-related direction made to an artist, before any action is taken, we’d ask the client to rephrase their direction descriptively. That’s when we would hear something more like, “make the yellow more intense and saturated, like, …twice as much as it is now.” Aha! And you know what the artist would do? They’d bump up that yellow’s intensity not by 2x but by 4x. And then they’d eyeball it and see if they’d need to do even more in order to give the perception that the color intensity was doubled.
Just like the essay writer’s example above, “Why do two lightbulbs not seem twice as bright as one?”,
the second lightbulb doesn’t appear to double the brightness of light in a room because our eyes process changes in lumination logarithmically. So, just as we’d need four lights of equal lumination to perceive double the brightness, we’d logarithmically bump up the intensity of the yellow to have it appear to be ‘twice as yellow” as it was before. Hence, bumping it up by (at least) 4x.
As it is thought by a cat that’s sleeping with their paw over their eyes so you can’t see them cuz they can’t see you- – but, yeah no, I SEE YOU CAT – a logarithm’s power already exists within you. It’s ready to be used as a way to bring compactness to an overwhelming situation and it exists as an equally powerful component to all those ‘quick growth’ seekers. Sure, you may need to knock down some existing biases or provide more explanation behind the benefits of this power, but then that’s the opportunity for you to own the very definition of your identity and of who you are.
Thanks math, you’re the best.