0) If you go into a supermarket, you will typically see a number of checkout counters, one of which is labeled "Express Lane" for x packages or fewer. If you make observations on x, you'll find it varies a good deal. In my home town, the A&P allow 6 items; the Shop-Rite, 8; and Kings, 10. I've seen numbers vary from 5 to 15 across the country. If the numbers vary that much, then we obviously don't understand what the correct number should be. How many packages should be allowed in an express line? (Pollak, 1987, pp. 260-261)
1) Answers to the homework
2) Quick review of exponential equations:
f(x)=ab^x
f(x)=a(1+- r)^x
$1 at 100% interest
multiple interest compounding
f(x)=P(1+r/n)^(nt)
every minute every second
f(x)=Pe^rt
Towns grow like this. All populations grow like this. What can we do with that information?
town of Bennington.
in 2010, Bennington had a population of 15,764. It has had a historical growth rate of 2.4%
in 2010, manchester had a population of 4,379. It has had a historical growth rate of 3.6%
radioactive plutonium!
Q(t) = 16(.5)^(t/24,100)
what would a function like this look like? what do each of the numbers represent for us?
car prices tend to depreciate exponentially. I bought a car for $9,000 as it was 4 years old. 6 years later, I sold it for $2,200. How much was the car worth (roughly) when it was bought? When will it be worth $1000?
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Of course, we're going to need a different tool to help use here -- logarithms.
2^x = 8
Some equations like this are doable in our minds
4^x = 8
Some we can change the bases to equal each other
2^x = 6
Some we are going to need another tool to use
What does 2^x look like?
What does log_2(x) look like?
Inverses of each other -- undo each other
How does this help us?
https://xkcd.com/1162/
https://www.wikiwand.com/en/Energy_density
http://en.wikipedia.org/wiki/File:Transistor_Count_and_Moore%27s_Law_-_2011.svg
http://earthquake.usgs.gov/earthquakes/
https://www.wikiwand.com/en/Richter_magnitude_scale#/Examples
http://www.geo.mtu.edu/UPSeis/magnitude.html
https://www.wikiwand.com/en/PH
http://antoine.frostburg.edu/chem/senese/101/acidbase/faq/what-is-pH.shtml
https://www.wikiwand.com/en/PH#/Applications
Logarithms are set up so we can find the exponent that will solve a problem for us. They make large numbers easier to handle.