Get some rhythm – Some LOGARITHM!

Logarithms are a frustrating function – the inverse of exponential functions, they are infuriatingly irritating to deal with.  But logarithms are a critical part of applied mathematics, and have been a critical tool in the mathematical kit since the dawn of mathematics (Which was when Ugg the caveman wondered how many rocks he had).

Logarithms are what you get when you find what power you must raise ten to to get another number.

The most fascinating part of history is the role of logarithms in the development of arithmetic methods. The simplest aspect is that it is simpler (and less error prone) to add instead of multiplying, so if you find the logarithm of two different numbers, the product of those numbers is equal to the sum of the logarithms. This meant, that in the age before calculators or computers, the process of multiplication was sped up and simplified by using a table of logarithms to do all calculations.

But logarithms are more than just an outdated way of speeding up arithmetic calculations – they are an important tool in graphing. By using a logarithmic scale on one or more axes, you can produce graphs that cover a broad range of numbers by compressing the axis. The graph to the left is an example of this – in one simple picture it presents the entirety of the universe – if this were linear scale, either the small items would be invisible, or the the graph would have to be so large that it is useless.

But the power of logarithmic algebra is the fact that logarithmic and exponential functions are inverses of each other. This means that each can be used to “undo” each other and arrive at a numerical solution for an expression. Equations such as exponential growth or decay, or logarithmic power can be reduced to solvable forms.

A review of the rules of logarithms can be found at this website; you should make yourself familiar with them, and transfer them to your bound reference – you never know when you might need them.

You will be expected to graph and interpret logarithmic functions; while your text book gives you some great practice activities, my favoured site for explanations of mathematical concepts is PurpleMaths – here is the link to their section on graphing logarithmic functions.

I have previously written a companion post to this one – a section on exponential functions and their relations to other interesting mathematical patterns.

You’ll also need to check out the following links (some of them require free registration). These websites give excellent support and may help you understand concepts “explained” in your text. (2, 3).

Since the school rules have changed to allow you to use your mp3 players in class, you might as well make them productive – try these podcasts: Math According to Mike, MathCast Central and Math Analysis.

Finally, a video to scar you emotionally so you can never forget logarithms again:

And two more that are slightly more helpful – if less memorable (1, 2)

Supporting notes are here!

See you in class.

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