Okay, I haven’t actually seen the latest installment in the *Star Wars* universe, so I’m merely assuming that the Dark Side of the Force is still an operative concept, but that’s not really what this post is about, anyway. What I’ve actually been pondering is the significance of the logarithmic function to the world of consumer goods, using some recently purchased dark beers as my example.

As a teacher of physical science, I have more than a passing acquaintance with mathematical functions and their relevance to the natural world, and part of my job is to point out to students the ways in which the various functions that they learn about in their Algebra II and Pre-Calculus classes can be used as the solution to specific types of problems. A decaying exponential function, for example, can be used to represent a system that is governed by a well-defined probability, but is only one of a number of functions that has a horizontal asymptote–you can’t just throw out the word “exponential” when the actual mathematical relationship is an inverse proportion, since the numerical predictions will not be in exact agreement.

A logarithmic function, like the one shown in the graph at right, has the property that the output value (on the vertical axis) is a continuously increasing function of the input value (on the horizontal axis). The specific property of the log function is that the output goes up by the same amount (say by one unit of whatever is being measured) when the input is increased by multiplying by a constant factor. (It happens to be a factor of two on this particular graph that I lifted from images.tutorcircle.com.) As a result, the increase in output value gets smaller and smaller for the same additive increase in the input value. Put another way, the amount you need to add to the input gets larger and larger for the same increase in output.

Where I’m going with this is to put the quality of some consumer good (admittedly a subjective measurement) on the vertical axis as a function of cost on the horizontal axis. In very broad terms, it seems to me that this relationship holds up pretty well for a wide range of products: spend *x* dollars and you get a certain level of quality; spend 2*x* dollars and you get a clear increase; however, to get the same kind of increase again, you need to go to 4*x* dollars, and so on from there.

What this also means is that for most of us there is some point of diminishing returns, either because of our spending priorities or because we don’t pay close enough attention to really be aware of the difference. If I try to really push this analysis, the log function indicates both an absolute minimum cost and an absolute minimum quality at the point where the graph crosses the horizontal axis. The graph goes up pretty rapidly for that first factor of two, but after that the gains quickly moderate, so it’s almost always worthwhile, if you can afford it, to spend something around two to three times the absolute minimum cost in order to get the best “bang for your buck.” (I can’t believe that I’m the first person to think about this relationship, but a quick Google search didn’t turn up anything obvious using the keywords *cost, quality, *and *logarithmic*, so I may have to speak to an economist to explore the idea further.)

My case in point here is a series of three imperial stouts. In this post from a couple years ago, I noted that a mass market American lager can retail as low as $0.75 for a 12 oz. serving, whereas the price point for a craft brew at your neighborhood liquor store typically starts at around $1.25 – $1.50 per serving. All of the beers I’m considering in this post are imperial stouts, which tends to push the price point even a little higher since their production involves extra ingredients and a higher alcohol content (typically somewhere in the neighborhood of 10% ABV).

The first and least expensive was an old favorite, the Black Chocolate Stout from Brooklyn Brewery. This is one of my favorite seasonal beers and it was on sale at around $8.00 for a four-pack at one of my usual stores, so I picked up a couple to get me through the winter. At $2.00 per serving this is a small extravagance, but definitely worth it to have around the house when I’m in the mood for something chewy and flavorful on a cold winter evening.

Moving up the cost ladder, we have Oskar Blues Ten Fidy, the name referring to the 10.50% ABV for what their website describes as a “titanic, immensely viscous stout.” This one came in at something around $12 for a four-pack of cans, so we’re now up to around $3.00 per serving, which is getting close to the point of diminishing returns. Definitely a very nice beer, but aside from the entertainment value of the label on the can, I’m not really sure that I can appreciate the difference enough to make it worth the expense for an everyday beer.

The final beer for this exercise is another Brooklyn product, a big bottle called Black Ops. In keeping with the name’s allusion to deep, dark secrets, any mentions of this beer on the Brooklyn web site are cryptic and humorously deny the existence of any such bourbon-barrel-aged imperial stout, but this was a feature offering at one of my local stores so I swallowed hard and forked over something in excess of $20 for a 750 mL bottle to share over dinner with my beer buddy Philip during an evening that also included board games with the teenagers. If we’re being generous we could say that 750 mL counts as three servings, but even stretching it that far we have more than doubled the price per serving again, and although this was a really fascinating mix of flavors, I wouldn’t choose to spend this much on a regular basis. I may watch for it rolling around again, but I’m not going to stock up and would only save a bottle for another special occasion.

But that’s alright, since we seem to have gotten past the season for heavy, dark beers, and just like in the movies, the light eventually wins out over the forces of darkness. So with the equinox and warmer weather rapidly approaching, it’s time to start looking for some spring seasonal beers, though there are a still a couple of prior season interlopers waiting for attention in the cellar that might see daylight before the end of Spring Break. Cheers!