# 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.

## Algorithm

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).

## Logarithms

**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 and hurricanes, for 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:

WebMD | HubPages | |
---|---|---|

website authority | 100 | 5 |

lifetime views/life length | 25000/250 (months) | 30000/2.5 (months) |

longterm value | 100 | 100 |

reader value (backlinks) | 10000 | 30 |

reader value (likes -Facebook, Twitter,...) | 30 | 10000 |

## Etymology Sidebar

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 9^{th}-century Persian astronomer- mathematician *Al-Kh(u)wa**hrizmi* ; 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.

## Comments

**MC** on May 20, 2016:

What's the difference. Pretend you are asked the question on an elevator, and would receive a million dollars, if the people understood it.

**Aficionada (author)** from Indiana, USA on June 09, 2011:

Thank you, Hugh Williamson and BlissfulWriter! I love seeing mathematical relations between all kinds of things in the world around us - but I know that I don't always (or perhaps don't often) understand what I see.

I hope that we don't become so accustomed to hearing about the Google algorithm that we start to think it's a bad word and/or ignore other algorithms that are also important. :)

**BlissfulWriter** on June 09, 2011:

I just know that earthquake magnitude scale is a logarithic scale. And that Google has an algorithm.

**Hugh Williamson** from Northeast USA on June 08, 2011:

I think our world is much more mathematical than we realize and your Hub makes the point nicely. I don't know how well a math themed Hub does on HP, but I for one enjoyed this.

Nice piece, thanks for writing.

**Aficionada (author)** from Indiana, USA on May 31, 2011:

"I was always intrigued at how every number could be expressed so differently simply by changing its base." - Isn't that the truth! I have to laugh at myself when I realize I feel so amazed that anyone could have a base 12 monetary system... until I realize that I still enjoy using a primarily base 12 measuring system. I like feet and yards, pints, quarts, pecks, etc.

I'm glad you brought up the binary system and computers. I have considered writing a sort of philosophical ramble on the type of thought process that reduces everything to only two choices, yes and no. Break something down into its simplest form, then - if done in proper sequence - all kinds of complexity can be achieved.

Thanks for reading!

**Xavier Nathan** from Isle of Man on May 31, 2011:

Bravo! You have done a great job explaining both of these mathematical concepts. It is not so long ago when we in Ireland and England used base 12 for our money and I remember as boy when we went metric. I was always intrigued at how every number could be expressed so differently simply by changing its base. Did you know that computers use base 2 and there is very good reason for this. 0=no and 1=Yes, it makes sense doesn't it?

I still think you would have made a great Maths teacher! Voted up.

**Aficionada (author)** from Indiana, USA on May 30, 2011:

Ahhh... Dolores. Hm. Thanks for reading! I think it's time for me to do a little bit of a rewrite of this and I'll try to pass word along when it's ready.

**Dolores Monet** from East Coast, United States on May 29, 2011:

If you did not do well in explaining this to some folks, remember that some people have to read something like this a dozen times to digest it. There are people who, well it's like reciting poetry to a brick wall. It's not you. It's me. haha. Thanks for trying.

**Aficionada (author)** from Indiana, USA on May 09, 2011:

Thanks, RTalloni... I think :).

I had hoped to try to make the words sort of clear, or at least as clear as my own understanding is... but if I didn't do very well in explaining it, that probably means I don't understand them as well as I hoped that I did! :)

**RTalloni** on May 09, 2011:

Well, I understood the last paragraph and the Etymology Sidebar.

Great stuff, really, I think.

I'm glad I have it in my back pocket, anyway. :)

**Aficionada (author)** from Indiana, USA on May 05, 2011:

Ahhhhh, jpcmc... So sorry to bring back bad memories... horrible memories.... :) Thanks for your kind words and yes, absolutely, you are more than welcome to put the words back in the mothball-fill attic. That's where I keep them! :) Cheers!

**JP Carlos** from Quezon CIty, Phlippines on May 05, 2011:

I had nose bleeds and migraines from all of these in high school and college. Memories flooded back...horrible memories. :)

What's great about this is now many people have an idea on the relationship between algorithms and search engine. ranking. Really informative. Thanks for the review. Now, allow me put it back in the mothball-filed attic. :)