# What is the advantage of a logarithmic model?

Mar 5, 2016

There are two main advantages: linearization and ease of computation / comparation, the former of which ties into the second.

#### Explanation:

The easier one to explain is the ease of computation / comparation.

The logarithmic system I think that's simple to explain is the pH model, which most people are at least vaguely aware, you see, the p in pH is actually a mathematical code for "minus log of", so pH is actually $- \log \left[H\right]$

And this is useful because in water, the H, or concentration of free protons (the more around, the more acidic), usually varies between $1 M$ and ${10}^{-} 14 M$, where $M$ is shorthand for mol/L, the appropriate unit of measurement, and yet, if we take the log the scale goes from $0$ to $- 14$, ( since we like to work with positive numbers we multiply by minus one, but that's besides the point)

Even though we lost the basic intuition we had with the original scale (where we know that, for example $1 M$ is twice more acidic than $0.5 M$) we're now working with a range easier to work with, not to mention that at least this particular systems works because usually we don't need the intuition we lost while doing this.

And that also helps with the first part, because you see, sometimes stuff in nature works exponentially, like for example, one type of analysis you might find in a chemical laboratory would look like this with raw data:

graph{10^(-x+2)+2 [-0.21, 19.79, -0.12, 9.88]}

But as soon as you take the log of it, it comes out more like

graph{x-2 [-0.21, 19.79, -0.12, 9.88]}

And the thing is, we can and like working with lines much more than that other curve, the line can be more easily manipulated, you can interpolate data much more easily, it's just simpler for the poor researchers to take the log.