Intro Question: Skeleton Tower.
A picture is up on the board of a skeleton tower, with questions below it.
https://www.illustrativemathematics.org/content-standards/tasks/75
Some reading to consider: http://hplengr.engr.wisc.edu/Math_Schoenfeld.pdf -- pages 60 - 62 (really the whole thing).
Another Problem to Consider: http://www.web-games-online.com/towers-of-hanoi/index.php
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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?
this is how carbon dating works.
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.