Logarithm: Compound Interest and End
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
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