I was playing the other day with some equations and thinking about how I was taught the logarithmic function back in school and how we usually think about it these days. And correct me if I am wrong but I can't remember a case when this concept was introduced in a some kind of visual way. The most visual thing I've ever seen about it was a graph of the logarithmic function – the well known curve. And may be you would say, this is because there is nothing visual about the logarithmic function per se. May be it is a purely mathematical concept, having nothing to do with visualizations.
I am kind of visual person, and if I can visualize a concept I feel much more comfortable playing with it. Space has some magic appeal that makes concepts look real and truthful. But there are things that cannot be visualized, you may say. In particular the nature of the logarithmic function. Well may be this is so, and in this case I shall end this blog post right here, but for the sake of argument let's just pretend that there is a visual way to see the logarithmic function. If so where could we try to find it?
So first of all let's acknowledge that logarithmic function is a reverse concept of the exponentiation, or a number raised to a power, which is also another concept rarely seen in visual way in modern days. But in the past, in particular in ancient Greece raising a number to a power has been seen as a purely geometrical concept, that's way to these days we call the second power, square and the third power cube. Interesting enough we use these geometric names even though they apply only for the two of the possible infinite number of powers. I don't remember anybody calling 4th power tesseract.
So know if we try to follow this ancient tradition visualizing the logarithmic function becomes quite easy. Let's see that on the example of a simple equation.
2^{3} = 8 => log_{2}(8) = 3
Now if a visualization for the for the number 2 representing a line of a cube raised to the third power give us the volume of the three dimensional shape well known to us as cube.
Then we can see the logarithmic function as follow: given a particular base side (2) and a shape volume (8) how many dimensions would we need.
And here right away comes the objection that the result of the logarithmic function can be a fraction while we clearly see the dimensions as distinct numbers. And that can mean that either the suggested by me analogy is inadequate or that our common sense concept of the nature of dimensions is rather naive. I am not sure which is true... Another objection would be that the half way between the square of 2 or 4 and cube of 2 or 8 is 6 but logarithm of 6 with base 2 is not 2.5. This is because 2.5 would not mean 2 dimensions and the half of the third dimension but rather half way on the way from two dimensions to three dimensions.
Anyway even if there is nothing in this visualization that would describe the real world in any useful way it might still be a good mnemonic trick for remembering what each part of the logarithmic function stands for in regards to raising a number to a power, or otherwise you would have wasted your time reading this blog entry :)