Algorithms and Logarithms: a User's Guide
The Google Panda update resuscitated a couple of mathematical terms that some writers had consigned to a mothball-filled attic: algorithm and logarithm. For many non-mathematicians, the notion of logarithms was a distant memory associated with sliderules and scientific calculators. A log table meant something to us way-back-when, but the trick is to remember exactly what … and why it should have anything to do with pagerank or SEO.
The term algorithm may not have been quite as familiar to some of us, but since it has been tossed around furiously since March 2011, it is worth looking at just exactly what it means, as well as why it is not a logarithm and why a logarithm is not an algorithm. All of the explanations that follow here should be understood as a layman's very rough paraphrase. More detailed explanations from any mathematician who cares to comment are more than welcome and may be incorporated in future updates to this article.
First of all, an algorithm actually can include or incorporate a logarithm. An algorithm is essentially a formula or a series of steps used to find the solution to a (math) problem. In talking about SEO results, people generally use the term algorithm as shorthand to talk about Google's process of finding various factors relating to an article, weighting them (mathematically) according to their importance, multiplying or adding them together (or subtracting and dividing, for that matter) according to some mysterious mumbo-jumbo, and finally reaching a manageable number that can be used as a quick way of ranking how visible to the search engine Google "wants" that article to be - according to the factors included in the algorithm. Other fields of study would use their own algorithms for their own purposes. And... an algorithm can be as simple as the process of finding the least common denominator within a group of fractions (an example taken straight out of the Random House Dictionary of the English Language).
A logarithm is related to exponents - raising numbers to higher powers, such as squared numbers and beyond. Baldly stated, the logarithm is actually the exponent itself. So, if the number 10 is to be raised to some power (X) so as to equal 10,000, the X (the power) would be 4.
10 to the 4th power = 10,000 OR
10 x 10 x 10 x 10 = 10,000.
The logarithm to base 10 of 10,000 is 4.
* * * * * * * * * *
But 4 is also the log of 16 to base 2. That is,
2 x 2 x 2 x 2 = 16; OR
2 to the 4th power = 16.
* * * * * * * * * *
So, saying that 4 is "the log" (logarithm) of another number is relatively meaningless, unless you know what the base is. Again, compare:
log (base 10) of 10,000 = 4 and
log (base 2) of 16 = 4
Between 10,000 and 16, there is definitely a huge difference!
* * * * * * * * * *
Let's look at one more comparison; this time, instead of looking at two logs that are the same, let's look side-by-side at two different logs of two different bases that give the same result.
log (base 10) of 100 = 2 and
log (base 2) of 100 ~ 6.5
* * * * * * * * * *
This comparison only intends to show that you can arrive at the same number by various routes – and if you are comparing the route only, then you don't have enough information. In order to understand a logarithm adequately, you need to know the base.
Certain scales in science are logarithmic, for example, the Richter scale for measuring earthquakes. That is, a 2.0 magnitude earthquake is 10 times (not double) the power of a 1.0 earthquake. Some scales used to measure powerful winds (tornadoes example) are logarithmic or roughly logarithmic.
More Examples of Algos & Logs
We actually use various algorithms and/or logarithms every day, probably without being aware that we do. As a rather silly everyday example, take the computer solitaire game "Spider."
In playing games of this type, the sequence of steps can be very important. It can make a world of difference whether you move one specific stack of cards before another and whether you turn over a new card in this stack as opposed to that one. In terms of this game, the steps you follow in a particular sequence would be the algorithm for the game.
If you wanted to go all mathematical, you might try to calculate how many different sequences of steps are possible. You can probably imagine that a game that includes over a hundred cards could create hundreds or thousands of patterns and sequences of steps to follow. The number might be so huge that you would want to express it as a base number with an exponent. That exponent would be the logarithm.
Simply put, the logarithm is one way to express a single, usually large, number. An algorithm is one way to express the steps used to find some result.
Algorithms and Logarithms Intertwined
In terms of the Google search engine rankings, some parts of the algorithm – the formula used to find the answer to a math problem – may in some instances be expressed as logarithms. No one actually knows what the "Google algorithm" is. But if someone like me were to take a guess, it would include factors like the relative authority of the website where the article is found; the lifetime number of views compared with the length of time it has been around; maybe some ranking of the long-term value of the content (vs. faddish or seasonal content); and how valuable the readers consider the material to be (which is probably what backlinks and social networking "likes" could help to measure). Numbers could be assigned to all of these factors; and the crucial element in all of this is the weight that would be given to each factor. Would or should the authority of the website count for more than its popularity?
These factors are just a token representation of the possibilities in the Google algorithm. As a matter of fact, around 200 different factors are actually included in it. On a really boring day, you might see if you can come up with a list of 200 different factors that could be used to evaluate a webpage. But in the meantime, the "algorithm" part refers to the way all of this information is collected, weighted (multiplied by some unknown factor), and then added or multiplied together to come up with a final result.
Just for the sake of example, here is a totally crass example of my own, made up (i.e. invented) just for this occasion. It is a comparison of a pretend article on liver cancer found on WebMD, but theoretically viewed only rarely, with another article on the same topic (also theoretical) written for a recent HubMob on HubPages which, against all odds, achieved incredible, instantaneous traffic.
The fictitious numerical results for this are:
lifetime views/life length
reader value (backlinks)
reader value (likes -Facebook, Twitter,...)
Since the two words are actually anagrams of one another, it is easy to assume they have similar derivations or etymologies. Not so – or, not entirely so.
Logarithm came into English from Greek by way of the Neo-Latin word logarithm(us). Greek log(os) meant word or speech and arithm(os) meant "number," as you can see in the word arithmetic. So, somewhere in the very distant past, a logorithm was a word that was used to express a number – or something like that.
Algorithm, on the other hand, was or is also known in English as algorism. It too had a pathway that led through Latin (Medieval Latin, in this case), but its starting point was Arabic, not Greek. It was actually the Latinized surname of the 9th-century Persian astronomer- mathematician Al-Kh(u)wahrizmi ; his name meant "the one from Khiva" (an area that is now in Uzbekistan and Turkmenistan; also the name of a city there). In 825 C.E. he wrote a treatise in Arabic "On Calculation with Hindu Numerals." When it was translated into Latin about 300 years later, his name was Latinized and included as part of the title (Algoritmi de numero Indorum) which presumably meant "by Al-Kh(u)wahrizmi, concerning the numbering system of the Indians." But, because of a quirk of the Latin language, it could also mean "Algorisms [whatever they may be] from the numbering system of the Indians." And so the word algorism was born, and became modified into algorithm by association with the Greek word arithm(os).
But maybe that change is not a bad thing, because algorism can also refer to the Arabic numerals themselves which we use in modern mathematics, and it can also refer to the method of computing with those numerals, plus zero (i.e. arithmetic). So it's really kind of nice to have a separate word that means what we now mean by algorithm. It's just that that word is not logarithm.
Still operating in the mode of imagination, the algorithm would multiply the first three factors together (WMD=1,000,000; HP=6,000,000); then add together the two types of reader value and divide that result by 100 (WMD=100.3; HP=100.3); add the two results together and multiply again by the website authority number squared. The final results (WMD=10,001,003,000; HP=150,002,507.5) could then be converted into logarithmic form to create a final ranking number (WMD=10 plus some decimal places; HP=8.5 with some more decimal places). So,in this fictitious example, because the results of the algorithm are expressed as a logarithm to base 10, a rank of 10 has ten times as great a total numerical value as a rank of 9 does.
Of course, if the base is different from 10, then the pagerank numbers could represent numerical values that are either much closer together, or even much farther apart!
Even before the problematic Panda, some SEO experts had begun a voluntary, somewhat anecdotal experiment to help figure out the logarithm base that Google uses. The experimenters were/are essentially trying to work backwards from a known (page rank) logarithm to figure out what the base must be, since that could be important in knowing how best to optimize for the Google search engine. But, even if they did or do ever figure that out, it would have limited value unless they also knew the relative or weighted value of the 200-give-or-take-a-few factors in the algorithm.
So there you have it. An algorithm is a formula, a series of steps that will solve a mathematical problem. A logarithm is a number that expresses the exponent of a base to arrive at some given number. A logarithm can be part of an algorithm, in some instances. And, since certain steps need to be followed in calculating a logarithm, I suppose we can also say that there may be an algorithm for figuring out a logarithm.
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