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December 16, 2009

Logarithms

   Tazz is explaining logarithms to Noah.  Shall I give you an example? 

In a nut shell: What power do I have to raise 2 in order to get 64?  Or, 2x = 64, solve for x.  In this question is the basis of a logarithms.  The 2x = 64 can also be expressed, x = log2( 64 ) = 6, because 26 = 64.

Which is bigger: 264 or 1040 Classic pre-calc problem, and the problem Tazz is explaining to Noah.
To solve, we will use the identity ln( ax ) = x ln( a )
ln( 264 ) = 64 * ln( 2 )
ln( 1040 ) = 40 * ln( 10 )
ln( 2 ) ≈ 0.69
ln( 10 ) ≈ 2.3
64*0.69 = 44.16
40 * 2.3 = 92
44.16 < 92

Therefore, 264 < 1040
Computing 264 isn't enough to overflow a calculator with 20 digits.  But if you wanted to see if 31000 is larger then 4700 then using a logarithm is a better option. 

How many digits are in 31000?  Again... let's use a logarithm.  Log in base 10, or log10, will tell us how many digits are in a number.  Why?  Think about this.  103 = 10 * 10 * 10 = 1000--4 digits.  108= 100,000,000--9 digits.  So digits = %u230Alog10( n )%u230B + 1. 
10x = 31000
log( 10x ) = log( 31000 )
Use identity : log( ax) = x log( a )
x * log( 10 ) = 1000 * log( 3 )
log( 10 ) = 1
x = 1000 * log( 3 )
log( 3 ) 0.4771
x = ( 1000 * 0.4771) = 477.13
%u230Ax%u230B + 1 = 478

So 31000 has 478 digits.  A more general formula is for the digits in ax is d = %u230Aa * log10(  x  )%u230B + 1

One last problem, which I've had to do: what is 3.142.71?  .
x = 3.142.71
ln( x ) = ln( 3.142.71 )
Use identity : ln( ax) = x ln( a )
ln( x ) = 2.71 * ln( 3.14 )
x = e2.71 * ln( 3.14 )
ln( 3.14 ) = 1.442
x = e2.71 * 1.442 = 22.22
If you are clever, you may have asked "That's great, but how do you calculate ln and e?".  Turns out that this isn't all that hard, but that's for an other article.
Magic Players

Magic Players

   I invested in more heating equipment in an attempt to keep warm in the basement.  Among the items was a heater you place your feet on.  It doesn't draw much power--less then 100 watts--but it pretty nice.  For me, as with many people, if my feet get cold, I feel cold.  So this is a big help.
   It is quite cold outside, and my basement is a bit on the drafty side.  I covered the windows with plastic--all but one was boarded up anyway, but that didn't seem to help much.  I noticed when walking up the stairs that the stairwell was the coldest place in the basement.  One day I noticed I felt a steady breeze.  Somehow, there is a connection to the outside world in the stairwell.  When the wind blows outside, it blows down here too.  Time to get a can of expanding foam and knock this problem out :)
Squeak

Squeak

   This is Squeak.  Sadly, Squeek wasn't alive much longer after this picture.  We knew we had mice shortly after moving in, and made some preparations.  Among them were adhesive traps and poison pellets.  While talking in the kiction with my roommate, Squeak started walking across the floor.  I tossed down some bread crumbs, but she wasn't too interested.  But she also didn't run away.  In fact, she seemed completely oblivious to our presents, even when I scooped her up in a jar.  Squeak was twitching and would often roll when trying to walk.  My guess is that she has eaten the poison pellets, and we were seeing the effects of nerve damage.  It seemed a terrible death, so Squeak was euthanized shortly after this picture was taken to spare her any more suffering.
   The aloe plant has moved from my work to the basement.  I don't have a light in my cube, so I'm going to have to get one if I want to talk my aloe back to work.  It needed a bigger pot anyway.  So it moved in with the chive.  The chive was doing really bad because of the move.  It was sitting upstairs, but not getting enough light at all.  It is coming back pretty good though.  Several new sprouts that are standing straight up.  So that's a good sign.  Nice thing about that which is alive--it likes to stay that way.  Good things when you have no gardening skills :)
   Someone doesn't know how to drive in the snow.  The last couple days have been a bit of a challenge.  My tires are all in need of replacement, and Eve isn't very heavy.  So there are some places were I need a running start so I don't get stuck.  So far she's done really well.  Takes a while to get up to speed, but I'm never in a hurry when the roads are like this.  The only problem I do have is one I can't really control--people love to pull in front of me.  Perhaps they are linked to that terrorist deer that took out my truck (*ponders*).
   Now it's winter!  We had a good snow storm move in last night.  So good in fact, Rockwell sent everyone home.  I hadn't been there an hour before the announcement came over the intercom to evacuate the building.  At first I thought maybe they broke a gas line or something, but it turns out the first part of the announcement was canned.  The snow plows were giving up trying to keep the parking lot and roads open.  There were high winds, and everything was drifting shut.
   I think this picture rather captures the state of the outdoors at the time.