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Old 04-28-2004, 02:31 AM   #23 (permalink)
Rangsk
Crazy
 
Location: San Diego, CA
I know I went on a rant about not needing this, but I figured it couldn't hurt to let you know.

A VERY common use of logarithms:

Logarithmic scales!

For example:
The Richter scale for earthquakes
The PH scale in chemistry
db scales for sound
Warp Speed (ok, ok, so Star Trek trivia isn't really that useful, but it's fun!)

All of these scales use log base 10. This means that every point is 10 times as large as the previous point. This is because you take the "actual" value and take the log base 10 of it.

So, let's make up a scale where logarithms would be useful- say, how large a living organism is.

We want the smallest organism to be of size 1, but we don't want humans to end up being a huge size... we want something that's down to earth. So, we'll use a base 10 logarithmic scale.

Let's say the smallest creature is the Rhinovirus, which is 2*10^-8 meters. We'll also say a human is about 2 meters. So, now we need to scale these numbers so that the Rhinovirus starts around 1 - we'll use a scale of nanometers. So, that means a Rhinovirus is 20 nanometers, and a human is 2*10^9 nanometers.

Now all we have to do is take the log base 10 of these numbers, and we have an easier to understand scale (we'll call it Rangskians, named after its inventor, me )

log_10(20) = 1.3 Rangskians
log_10(2*10^9) = 9.3 Rangskians

Now, why is this scale easier to see how large something is? Well, first it will always deal with relatively small numbers. If a virus is around 1 and a human is around 9, then they are numbers that are easy to deal with. Second, you can infer a lot about the numbers. 9.3-1.3 is 8, which means that humans are 10^8 times larger than a virus.

The moral of the story? Logarithms will allow you to take large numbers and reduce them in size drastically.

As for equations that use logarithms, I dare you to solve this equation for y without them:

x^y = 50

(Here's how to solve it)
log(x^y) = 50
ylog(x) = 50
y = 50 / log(x)

You can now graph all possible solutions easily.

As for dervitives, I use them so much, I can't remember how I did math without them. Trust me, you will never regret doing those hundreds of derivitive problems they will make you do in your first calc class.
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