Why you should love logarithms

A problem involving factoring and logarithms

I was recently in the midst of a pitched battle, waging war against the forces of ignorance  with the weapon of algebra, when The Battlefield (who is 14), opened his unenthusiastic eyes and asked me why he needs to know how to factor polynomials.
I have taught algebra in the high school and college setting. I have tutored people in middle school and helped people study for the GRE. I have fielded the question of “Why do we need to know this?” hundreds of times. The answer always falls into three categories: because it’s on the test, because it’s good for you, because if you learn it you can go learn calculus.
When you are The Battlefield, and as determined to never need much math in adulthood as a landmass could be, being able to recognize the difference of two squares is not your problem. It is an age-old conflict between the forces of chaos and the forces of order. In my endless engagement with the dark morass of ignorance, I foolishly persist in adding another reason to the three listed above: because it’s cool. Obviously, I am an over-educated idiot.
Some educated adults feel free to express disdain for topics in math that they once found baffling, and logarithms is a common enemy for these folks. The logarithm is actually a very handy thing, invented at a time before people understood exponents, back when long division had to be done by hand.  The history of logarithms is a very interesting story, which I will have to tell another day.
I have in my repertoire a story I tell whenever the existence of logarithms requires justification (beyond the four reasons already mentioned).  
Imagine you are a marine biologist, I say,  And you have been asked to survey all the animals you find living in a specific cubic kilometer of  Hudson Bay and record their population size on a graph. What might you find? A pod of 9 Beluga whales,  perhaps a few more arctic cod and sculpin fleeing the hungry whales, but what if there was a bunch of zooplankton, where individuals are millimeters long but there are millions of individuals, or a single Bowhead whale, eating zooplankton? How would you record the numbers in a graph?
Even if you made things simple, such as whales 10, fish 100, zooplankton 100,000,000, you are going to have trouble showing that on a graph. The scale is going to be a problem, even if you have a really, really big piece of paper.
Common log, which is base 10, is a good way to show the magnitude of numbers, and in our example, the log of our population numbers yields whales 1, fish 2, and zooplankton 8. Yes, common log counts the number of zeros, and gives us data we can easily fit onto a small graph. Our only further responsibility is to ensure that we identify the scale as logarithmic. Hopefully our audience understands.

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