Logarithm: Compound Interest and End

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Here's a good example from engineering. An aerospace engineer might want to graph the lift of a wing versus the size of the wing, and he wants to show everything from an insect wing all the way up to a jumbo jet.

A fly's wing is maybe 0.1 inch long, and a jumbo jet wing might be 1000 inches long (about 80 ft). It would be pretty tough to put more than one insect on that graph - if you scale the wing length axis to fit on a sheet of paper, all the points for insect wings would be jammed up against one side.

Now, instead of plotting length, what if we plot the logarithm of length? There will be as much space on the graph between 0.1 inch and 1 inch as there is between 100 inches and 1000 inches, because

log(0.1) = -1
log(1) = 0

log(100) = 2
log(1000) = 3

So the graph will be much easier to read.

Logarithms are used in a lot of places to scale numbers when there's a big range between the smallest and the largest numbers of interest, which makes them easier to talk about.

y=yi x e^-kt
where:
y - different between temprature of body and the constant temp of room yi - initial temprature difference of body and room
e - eulers number (2.718...)
t - time in mins
k - constant for that particular body (usually what u are trying to find out in class tasks)

using logarithms, newtons law can predict how how a body (such as cup of coffee) will be after any given period of time.

Example 1: A $1,000 deposit is made at a bank that pays 12% compounded annually. How much will you have in your account at the end of 10 years?

Explanation and Solution:

At the end of the first year, you will have the $1,000 you had at the beginning of the year plus the interest on the $1,000 or . At the end of the year you will have . This can also be written . At the end of the second year, you will have the you had at the beginning of the year plus the 12% interest on the . At the end of the second year you will have

This can also be written . Another way of writing this is to write the balance at the end of the second year as .

At the end of the third year, you will have the you had at the beginning of the year plus the 12% interest on the . At the end of the third year you will have

This can also be written . Another way of writing this is to write the balance at the end of the third year as .

By now you should notice some common things in each end-of-year balance. For one thing, the exponent is the same as the year. The base is always 1 + rate or 1 + .12. The $1,000 will always stay the same in the formula. Now we can write the balance at the end of 10 years as which can be simplified to

rounded to $3,105.85.

Example 2: An $1,000 deposit is made at a bank that pays 12% compounded monthly. How much will you have in your account at the end of 10 years?

Explanation and Solution:

In this example the compounded is monthly, so the interest rate has to be converted to a monthly interest rate of . At the end of the first month, you will have the $1,000 you had at the beginning of the month plus the interest on the $1,000 or . At the end of the month you will have . This can also be written . At the end of the second month, you will have the you had at the beginning of the month plus the 1% interest on the . At the end of the second month you will have

This can also be written . Another way of writing this is to write the balance at the end of the second month as .

At the end of the third month, you will have the you had at the beginning of the month plus the 1% interest on the . At the end of the third month you will have

This can also be written . Another way of writing this is to write the balance at the end of the third month as .

By now you should notice some common things in each end-of-month balance. For one thing, the exponent is the same as the number of months that have passed. The base is always 1 + rate or 1 + .01. The...
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