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Transcript of Logarithm Prezi
By: Grant And Hunter
4 Real World Applications
The logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 101010 = 103.
You take the exponent off of the variable and put it in front of the Log
Dividing logarithms breaks into subtraction
Multiplication breaks into addition
Logarithms date back to 2000 BC in Babylon containing parts that may be in the logarithms we use today.
But in 1614 a book was written explaining logarithms written by John Napier. Another man named Joost Bürgi independently invented logarithms but published six years after Napier did.
Napier was born in Scotland
The first logarithm looked like this:
where the L stood for a "artificial number" but soon was given the name of Logarithm
Measuring Annual Interest
One of the many and varied applications of logarithms is the calculation of pH levels in varying substances. This entails calculating how many hydrogen molecules these atoms contains. An example of the equation used to find this is: Calculate the pH for a specific [H+]. Calculate pH given [H+] = 1.4 x 10-5 M
pH = log10[H+]
pH = log10(1.4 x 10-5)
pH = 4.85
As you can see this is very simple to do and allows the person to be able to find the acidity of all kinds of different chemicals.
Logarithms are utilized by scientist to calculate how powerful an earthquake is. They use something called the Richter scale which is measured by logarithms in order to be able to measure an earthquake's power. The equation used for this is: M=log , Where "I" is the intensity of the earthquake and S is the intensity of a standard earthquake whose amplitude is 1 micron =10-4 cm. So for instance when the Haiti earthquake happened, this is how the seismologist measured the intensity of it.
Measuring the Magnitude of Earthquakes
Measuring Decibel Level
When an acoustical engineer is attempting to calculate the loudness of sound in a certain space, he uses the decibel scale. This scales utilizes logarithms to measure the intensity of the loudness. The equation for this particular scale is: Decibels (dB) = 10 log(P /P ), where P /P represents the power ratio. Once someone knows this they know what safety precautions must be utilized in order to prevent permanent hearing damage for the various people that go to that building or room.
When somebody wants to calculate how much debt they have accumulated over the year, they use the PERT equation. This equation has a very specific use that it is very good at. The equation itself is: A=Pe . By utilizing this an average person can for instance calculate how much a loan on the new car that they are going to buy will cost.
r * t
Decay Factor (Exponential Decay)
y = a * b
Exponential growth is when the b factor from the exponential function is greater than one. It is used to model growth.
Exponential decay occurs when the "b" factor in an exponential function is greater than zero but less than one. It is used to model decay.
An asymptote is the line that a curve approaches until it reaches infinity.
The interest formula is A=Pe . The "A" variable is the amount of interest accumulated plus the original amount, the "P" variable is the amount of money that was originally used, "e" is just exponential, "r" is the interest rate, and "t" is time. An example of this is "A man has put his saving of $1000 into a bank that has a 10% interest rate. What is the total interest he will accumulate over a time period of 4 years?" To do this, just plug in the info like so:
A= ($1000)(e )
Change Of Base
A logarithm is a quantity that a number must be raised to produce a given number. The number itself is the base of the logarithm. For instance:
log (8) =3
A common logarithm is a logarithm with a base of ten. For example: log (10) = 1
The logarithm of a number is the exponent by which another value, the base of the log, must be raised to to produce that number.
log10(1000) = 3.
An exponential equation is an equation where a variable occurs within the exponent. An example is: 7 = 7
2x + 1 = 3x - 2
3 = x
Natural Logarithmic Function
A logarithmic equation is where there is a logarithm on both sides. For example:
log (x) = log (14)
A natural log is a log to the base of e.
For example: ln (x + 4) + ln (x - 2) = ln 7
ln (x + 4)(x - 2) = ln 7
eln (x + 4)(x - 2) = eln 7
(x + 4)(x - 2) = 7
x2 + 2x - 8 = 7
x2 + 2x - 15 = 0
(x - 3)(x + 5) = 0
x = 3 or x = -5
x = 3 checks, for ln 7 + ln 1 = ln 7.
x= 5 does not however, so x=3 is the only viable answer.