a) Read the information using the link below: https://www.mathsisfun.com/algebra/logarithms.html
b) Take the 10 questions quiz at the bottom of the page. Have the record emailed to yourself. (If I ask you to show it to me, you should be able to)
c) Bring a practical example of using inverses related to exponential and logarithm functions. (ex. compounded interest…etc)
Bests,
I can think of one use of logarithm and it is in binary. As everyone knows computer memory is using binary system. For example log base2 (n) tells us how many address bits you need for a memory of size (n). For instance, if you have memory of size 1024 then you have, 2^x=1024, log 2(1024) = 10. As seen logarithms might be very useful if you are working with computers.
That is a great example Norbert. Binary search algorithm is one of the most effective searching method in the computer science.
While linear search algorithm requires n comparisons, the binary search requires roughly log with base 2 of n. (in a sorted list though). So, binary search algorithm uses logs. This is the computer science concept, so you don’t need to worry if you don’t understand it, but it does use the logs.
Although logarithms may seem like they would not have much of an impact on today’s society that is actually far from the truth and can be used in many economic situations. In fact, Logs can be used to determine something as large as the increase in Americas GDP( gross domestic product) which is the monetary measure of all goods and services produced within America in a specified period of time. For example, in a 10-year span (2000-2010) the U.S GDP went from 9.9 trillion dollars to 14.4 trillion. Using the equation LN(14.4/9.9) we find an answer of .374 and with that, we need to divide by t (time) which would give us .374/10 = .0374 which times 100 would give a final percent of 3.74% in the span of 10 years.
We can see the use of logarithms in the evaluation of earthquakes, it allows scientists to determine how destructive the earthquake was. To determine the magnitude you need the Richter scale, its a base 10 logarithmic scale. its a ratio of the amplitude of the seismic waves to the minor amplitude M=logA/s. The A is the amplitude and the S is the standard earthquake. since the Richter scale is a base 10 each number increases 10 times stronger than the previous number on the scale.
The use of logarithms can be seen in the determination of how destructive is an earthquake. Scientists use the Richter scale to evaluate the intensity. The scale is a base 10 log, its the ration of the amplitude of the seismic wave and the standard, M=logA/S. Since the scale is base 10 each number increases by 10, that means the intensity is 10 times stronger than the previous.
The use of logarithms can be seen in the determination of how destructive is an earthquake. Scientists use the Richter scale to evaluate the intensity. The scale is a base 10 log, its the ration of the amplitude of the seismic wave and the standard, M=logA/S. Since the scale is base 10 each number increases by 10, that means the intensity is 10 times stronger than the previous.
The use of logarithms can be seen in the determination of how destructive is an earthquake. Scientists use the Richter scale to evaluate the intensity. The scale is a base 10 log, its the ration of the amplitude of the seismic wave and the standard, M=logA/S. Since the scale is base 10 each number increases by 10, that means the intensity is 10 times stronger than the previous.
Logarithms can be used in pH calculations. ph=-log10(H^x). pH measures how acidic or basic something is. For example, water contains 1*10^-7 moles so you can plug that into the log function to make pH=-log(1*10^-7) which after simplification would be 7. In this scenario, if you have the moles of a substance you can find its pH. So scientists can use log equations in this application to find where a substance lies on the pH scales.
Let’s assume you have $100 to open a savings account at XYZ Bank on January 1. you will you have $100,000 in five years? So LN(100,000/100)/5=? will give give you the percentage %138. The financial world often refers to compound interest as “magic” because it is one of the most fundamental ways to build wealth yet takes the least amount of effort. i find this cool “”Logarithm” is a word made up by Scottish mathematician John Napier (1550-1617), from the Greek word logos meaning “proportion, ratio or word” and arithmos meaning “number”, … which together makes “ratio-number” !”
logarithms are used in Sound. You can use logarithms to find dB (Decibel) which is a unit of measurement to find out how loud a acoustic and electronics can get. So to find the dB of 2 amplitude you will write 20log(2A/A)= 20log10(2)+20log10(A/A) which later equals to 6.02. So with the use of Logarithms people can adjust there equipment and make it sound better and make different type of sounds.
My personal favorite example of an exponential function is the concept of growing value in terms of the business world. For example, with houses. A person, for example, might have bought house (in a certain neighborhood) for around 500k in the 1990s and that house now is worth triple the price it initially cost. The value of the property increases over time (some may argue that it might not consistently increase, but just for simple understanding of the concept, the overall ending is that the value of the house increased). Another great example of exponential growth that is more stable is a Government bond. A person can “lend” so to speak money to the government and will have a certain set interest (a locked-in rate ) that gets added onto the amount they invested; working like an investment. For example a $2000 investment at a 3% rate , annually for 12 years would yield $2851.52 . Similarly in investing, inorder to find out how much you need to invest to have an out come of approx $2800 you would solve for the principal amount and get a value around $2000. I like examples with money because for me it’s understandable and a more tangible concept than examples that have to deal with architecture.
Logarithms are used a lot in chemistry, it can be used to calculate how acidic something is when we talking about “ph” or the calculation of grams of any elements after a certain amount of time. It can be really useful if you want to become a pharmacist or scientist and you want to calculate a certain medicine or substance half-life or if something is decaying or growing in a certain way with the combination of substances. Also if you like animals and you want to know the proximity of an animal population in any year from now.
Four years ago a forest was repopulated with a new species of birds. Then 100 specimens were introduced. Currently, it is estimated that there are 25,000 copies. It has been concluded that the number N of these birds is given by the formula N = (A)(e)^(Bt), where A and B are constant. The time t is considered expressed in years.
a) How long will it take to wait for 200,000 species of birds?
b) How many species will there be after ten years?
The mass of the Dog is 3*1.42^t pounds.
A. what is the mass of the Dog after 5 hours of eating?
B. when will the mass reach 12 pounds.
1. y(t)=3*1.42^t
2.y(5)=3*1.42^4=8.13.
3.log(4)/log(1.42)=t=3.95
Exponential Growth can be used to describe Moore’s Law. This law is the observation that the number of transistors in a dense Integrated Circuit doubles every two years. You can use logarithms to find out how many transistors are in a modern, 2019 CPU compared if given the number of transistors in a CPU from 1980.
For example: In 1980, the Intel 8088 had 29,000 transistors. If the number of transistors doubles each year, how many transistors will an Intel CPU in 2010 have?
logarithm is the inverse function to exponentiation (it is an example of a concave function). That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts repeated multiplication of the same factor
An Example of a Logarithm Function is
The mass of the Cat is 2*1.32^t pounds.
A. what is the mass of the Cat after 5 hours of eating?
B. when will the mass reach 12 pounds?
Logarithm function is used everywhere around us. It is used to calculate the exponential growth of things. For example, $1 can buy several things today in ten years you will be able to buy even fewer things. If inflation is about 2% each year, and the price of a pair of jeans is $40 now, the price of the same jeans x years from now will be: P = 40(1.02)^x. What will the jeans cost 10 years from now? P=40(1.02)^10 each jeans 10 years from now will cost approximately $48.76.
Thank you everyone for great examples of real life applications related to exponential and logarithmic functions.
How do you see these two type of functions as inverses of each other???
This would be a further question. Please reflect on this. You can reflect in your application. Just highlight it, so the others can see it.
The decibel system is used to measure sound level, but it’s also used in electronics, signals and communication.. It’s related to logs because suppose we have two loudspeakers, the first playing a sound with speaker S1, and another playing a louder version of the same sound with speaker S2, but everything else remains the same. The difference in decibels between them is shown as 10 log (S2/S1) dB , where the log is set to base 10. This allows you to use logs in order to find the decibels differences between one another.
You are welcomed to look and work on this desmos activity related to compound interest written by Prof Poirer.
Code: 8M247H.
If you invested $1,000 in an account paying an annual percentage rate compounded quarterly, and you wanted to have $2,500 in your account at the end of your investment time, what interest rate would you need if the investment time were 1 year, 10 years, 20 years, 100 years?
One way I can think of the logarithmic function being applied in real life is through business management. For example, a manager of a successful business wants to manage and collect information on their business’ stocks and investments. As a successful and big company, there would be many high values of income invested into it. If all stocks are in millions or above, the best way to collect that information is with the logarithmic function: logb(a)=x.
If their stock were in total average of 100k in 2012 that would be 10^5=100,000 as well as log 10(100000)=5
If it were now in total of 10m in 2015, it would become 10^7 as well as log10(10000000)=7
If a software company wants to know about the amount of usage of their application by people per year, the estimated growth of that usage and the amount of usage in a future year the use of logarithmic and exponential equations are very important.
For example:
A software company’s application had 8.4 million downloads in the year 2005 (initial year) and 13.6 million in the year 2010.
a. We have to figure out the number of downloads in the year 2015.
Basic exponential equation formula:
f(t)=c*(b)^t
f(t)=8.4 * (1.101)^t (the equation we use in this problem)
Since the year 2015 is 10 years away from the year 2005, we set t = 10
f(10) = 8.4 * (1.101)^10= 22 (approximately)
In 2015, the number of downloads is gonna reach approximately 22 million.
b) We have to figure out which year the number of downloads will reach 20 million.
f(t)=8.4 * (1.101)^t
20=8.4 * (1.101)^t
To solve for t variable we have to convert this exponential equation to a logarithmic equation. We find t=approximately 9 years. So, the application will reach the 20 million downloads landmark in the year 2014.
If you invest $9,000.00 at 4% interest, compounded annually, how much is the account at the end of 5 years ?
For this question we would have to use the exponential formula a=p(1+r)^t.
a=the final amount after a certain amount of years.
p=the present or initial value.
r= The rate in decimals
So a=9000(1+0.04)^5
a=$10,949.88 is how much the account is gonna have at the end of 5 years.