Logarithm (and Flogarithm)
The logarithm of a number is approximately the number of digits it has (the flogarithm), and it grows very slowly: the flogarithm of 1000 is 4, of 1 million is 7, of 1 billion is 10.
THE IDEA
A NOTE ON THE LOGARITHM, AND THE FLOGARITHM
It has come to my attention that hardly anybody knows what the logarithm is. Let me take a step toward fixing this. The logarithm of a positive number N, called log N, is the number of digits it has.
Wait, really? That’s it?
No. That’s not really it. We can call the number of digits the “fake logarithm,” or flogarithm. It’s close enough to the real thing to give the general idea of what the logarithm means in a context like this one. The flogarithm (whence also the logarithm) is a very slowly growing function indeed: the flogarithm of a thousand is 4, the flogarithm of a million, a thousand times greater, is 7, and the flogarithm of a billion is still only 10.*
When to use it
When estimating the order of magnitude of large numbers, the logarithm (or flogarithm) gives a quick sense of scale.
In computer science, logarithms are used to measure the size of data sets (e.g., binary search).
In finance, logarithms are used to calculate compound interest over time.
In science, the Richter scale for earthquakes is logarithmic.
FROM THEORY TO EVERYDAY LIFE
In practice
Flogarithm values
The flogarithm (number of digits) of 1000 is 4, of 1,000,000 is 7, and of 1,000,000,000 is 10.
A MOMENT TO REFLECT
Put it to work.
How many digits does the number 10^10 have? What is its flogarithm?
Why is the logarithm called a slowly growing function? Can you think of a function that grows even slower?
If the flogarithm of a number is 5, what can you say about the number?
CONTINUE THE CONVERSATION
Source texts
- How Not to Be Wrong The Power of Mathematical Thinking (Ellenberg, Jordan)