Extra Help (LOL)

Gotcha ya !! You thought LOL stood for "Laugh Out Loud".

It does in the email world, but in the Math Classroom it stands for Law of Logs !!

Today we were studying Logs, no, NOT the kind you use for firewood or to build a log cabin house. These logs are the operation (function) we use to find exponents.

Ok, i just lost half of my audience. With only 4 people reading these, there goes 2 people. Please stay. Try to follow for fun. I guarantee you it is an important concept and it DOES have real world applications (NOT like Algebra).

Subtraction is the opposite of addition, Division the opposite of multiplication, square root the inverse of squaring...(are your palms getting sweating ? sorry)

3 square is 9...and the square root of 9 is 3...got it ?

In Math we first learn to go forwards and then we learn to go backwards.

In Algebra we learn to "FOIL" (ex: (x+1)(x+2) = x^2 + 3x + 2 )....then we learn to go backwards which we call Factoring...(ex: x^2 + 3x + 2 = (x+1)(x+2) )

Of course this has become obsolete because our graphing calculator have a program that solves the equation for you, so we really do NOT need to factor any more. (But please do not tell all those old ladies still teaching factoring after 30 years. Hey, even I still have to teach factoring after all these years...what a waste of time.)

Now we get into Algebra 2 and we start to study exponential functions.

The thing is that in the real world things do not grow in a nice line. (y = mx+b)
And only a few things follow the parabola (y = x^2 )

One of my favorite formula's for a baseball or football traveling through the air or a cannon ball ( y = -16x^2 + velocity x + height ). In Algebra 1, we find the maximum height of the ball and how long does it take to hit the ground ? (in the real world we call that a field goal or a home run !!)

But in the real world (nature) things grow exponentially. Like the growth of trees and the more importantly the growth of MONEY.

A little review is necessary. When we are in middle school, we learn about "simple interest". If you put $1,000 in the bank and it grows at 6% a year, in 5 years you should have $1,300 in the bank. The original $1,000 you put plus the extra $300 we call interest earned. Mathematically it looks something like this:

A = P ( 1 + rt )....therefore A = $1,000 ( 1 + .06(5) ) = $1,300.

Now our boy Benjamin Franklin loved something called "compound interest". This means we "earn interest on the interest". That is a beautiful thing if you are a banker or if you are saving money. It is an ugly thing if you are borrowing money...lol.

Compound Interest is an exponential function. We say it grows exponentially FAST.

(Population growth also grows exponentially until come force of nature slows it down. We have experience exponential growth in the 1900's but now that the age limit of humans as stabilized at 75 or so, the growth has slow down especially in "developed" continents like America and Europe. In Europe they are paying couples to have children and in the USA they give you tax incentives to have children. Africa population has stabilized for other reasons (hunger and Aids), but Asia continues to add to the overall World population especially India and China, two countries that have over half of the entire world's population. China has tried to control this by state law allowing only one child per family.)

but i have digress...one last word on world population. It was predicted we would double from 6 Billion in 2000 to 12 Billion by 2020, but it looks like that will NOT happen. The new date is now 2050. There is even fear among some people that the world's population will actually start to decline after 2050. We will be the only animals the purposely will bring our species population into decline. And the fear is that once that decline starts, it will be difficult to stop it.

Now getting back to money.

What if we were to compound our money "yearly" (every year). That $1,000 we started with would be $1,060 after the first year. Now if we earn 6% on that extra $60 we earned then after two years we would have $1,060 plus $ 63.60 for a total of $1,123.60.

But that is too much work so we create a formula...and it goes something like this

A = P ( 1 + r/k )^(kt)....A = $1,000 ( 1 + (.06/1))^(1 * 5) = $1,338.23
($1,000 compounded annually or once a year for 5 years at 6%)

But if we wanted to compound it monthly then k = 12
A = ( 1 + (.06/12))^(12*5) = $1,348.85

How about daily then k = 360 (in the business world there are 30 days per month)
A = ( 1 + (.06/360))^(360*5) = $1,349.83

Now we get real greedy and we want to compound it every hour, every second,
continuously !!! (aha, the thought of infinity...we call that Calculus...Archimedese in 250 BC thought about infinity but that if for another blog)

( 1 + (1/x))^x where x goes to infinity is 2.728....and we call that "e" (after the greatest mathematician of all time Euler 1700's)

and now we get "Pert shampoo" for compounded continuously

A = P * e^(rt)....yes, Pert shampoo

A = $1,000 e^(.06 * 5) = $1,349.59

Now we have to go backwards...and start talking about logs.

Let me tell you a story about "log diving" in the springs near Gainesville, Florida.......

Some believe it was Benjamin Franklin who said, "Compound interest is the eighth wonder of the world," while attributing "Compound interest is the world's greatest discovery," to Albert Einstein. Still others attribute the whole thing to John Maynard Keynes.

Historical note: the Indains were given $24 of trinkets in the 1500's for Long Island. At 6% interest rate over 500 years it would come out to

A = $24 e^(.06 * 500) = 2 followed by 14 zeros ($200,000,000,000,000 = 200 Trillion)

COMMENTS:

(OK, the Extra Help blog just makes me nervous and feel stupid as I can't figure any of that stuff out. I should know how to do all that stuff, but I don't and never will no matter how much I try! Just one of my many, many deficits.

The autobio blog looks great! I'm really glad you got to Cuba over the Christmas holidays; it must have been very emotional, and I really look forward to reading the chapters.

Thanks for passing the links on.

This writer is a computer nerd...and HE feels nervous...lol)