Exponential Functions
In an exponential function, f(t) = P(1±r)t, this letter represents the "initial amount."
What is "P"?
Vanayja and Alzuri both tried to solve this logarithm: log28.
Vanayja got 4 and Alzuri got 3. Who is correct and who is incorrect? What do you think the person who got it incorrect did wrong in their problem solving?
Alzuri is correct and Vanayja is incorrect. Vanayja may have gotten her answer wrong because she solved the problem 2x=8 rather than 2x=8.
Imani and Kentton were trying to solve for x in the following equation: 2x=8. Imani got 3 and Kentton got 4. Who was incorrect in their solution for this problem? Why do you think they got it wrong? What mistake did they make?
Kentton got the answer wrong in this problem. One possible reason he got it wrong is because he accidentally tried to find the answer to 2x=8 rather than 2x=8. 2*4=8 but 24=2*2*2*2=16.
Imani got 3 because 23=2*2*2=8.
This is how you would say "log464" out loud.
What is "log base 4 of 64"?
Construct an exponential function to model the following statement.
A population of stray cats, known to double every year, starts at a population of 8 in 2008.
When solving for x in this logarithmic equation, log5x=5, Judaven got 25, Khaled got 3125, and Kandi got 1. Which group member got the answer correct and why?
Khaled got the answer correct. He got it correct because they were trying to solve the equation 55=x. 55=3125, therefore x=3125.
After constructing their exponential function, f(t)=8(2)t, Jamiyah and Ahries got started with plugging in t=4.
Jamiyah plugged in 4 to get f(4)=8(2)4=8(8)=16.
Ahries plugged in the same thing to get f(4)=8(2)4=8(16)=128.
Who evaluated incorrectly? What mistake did they make?
Jamiyah evaluated incorrectly. Instead of calculating 24 to be 2*2*2*2=16, she calculated it to be 2*4=8. She misinterpreted the exponent.
Evaluating this logarithm: log381, you would get...
What is 4?
Construct an exponential function for the following statement:
A handbag was purchased in 2009 for $16,500. This handbag model is known to increase in value at a rate of 8% every two years.
Additionally, what is our "t" if we're trying to find the value of the handbag in 2017?
What is f(t)=16,500(1.08)t and t=4?
When rewriting their logarithms as exponential equations, David and Gavin got different answers for one problem. The logarithm they were trying to rewrite was log2256. David rewrote the logarithm as 2x=256 and got x=8. Gavin rewrote the logarithm as x2=256 and got x=16. Who is correct and why?
David is correct because logarithms are in the format logba=x. To rewrite a logarithm into an exponential equation, it must be rewritten in the format bx=a. So, considering log2256, we see that the base is 2, therefore the base of the exponent would be 2, not x. Therefore, the correct rewrite would be 2x=256.
Noah and Luke were arguing on who was right in solving the exponent 83. Noah said the answer was 24. Luke said the answer was 11. All of a sudden Jamar came in late to class and said the answer is 512. Who is right? Why are the other two wrong?
Jamar is right. He saw 83 and must have known that it is equal to 8*8*8=512. Noah may have gotten it wrong because instead of multiplying 8 to itself 3 times, he did 8*3=24. Like may have gotten it wrong because he did 8+3=11.
If you rewrite this logarithm, log24 as an exponential equation in the format bx=a, it would be this.
What is 2x=4?
Construct a word problem for the following function and solve:
A car, purchased originally for $50,000 in 2010, is expected to decrease in value by a factor of 1/2 every ten years. How much is the car expected to be worth in 2040?
What is f(t)=50000(0.5)t, where t=3, therefore, the car will be worth $6,250 in 2040?
Marshawn and Irshad were solving an exponential word problem using logarithms. They constructed the function 2000(1.1)t=5000, and then attempted to solve for t.
Marshawn's solution involved dividing both sides by 2000 to get (1.1)t=2.5, then getting the log of both sides to get log(1.1)t=log(2.5).
Irshad's solution involved multiplying 2000 by 1.1 to get 2200t=5000, then getting the log of both sides to get log(2200)t=log(5000).
Who was on the right track? Why?
Marshawn was on the right track because he was following the order of operations. (1.1)t is its own exponent so you can't just multiply the base of the exponent to another number.
Darionna had an epiphany in the middle of class. She said "oh my god, I just realized 24=42." Is this a true statement or is Darionna just trippin'? How do you know?
This is a true statement. 24=2*2*2*2=16
42=4*4=16
Both exponents equal 16, therefore, they are equal.
What is "5"?
Construct a function for the following word problem and solve:
A certificate of deposit gains interest at a rate of 0.5% per month. If the balance was $10,500 in January 2025, what will the balance be in the certificate when it matures in January 2026?
What is f(t)=10500(1.005)t where t=12, therefore the balance will be $11,147.62 in January 2026?
DaMya and Dearra were in the final stages of solving their word problem and getting their final answer. The were at the point where their equation was (1.07)t=5 and needed to solve for t.
Dearra remembered that in a past lesson, we were able to bring down the t, so she suggested that we bring t to the front to get t*(1.07)=5, then divide both sides by 1.07 to isolate t. She ended up getting t=4.67. When she asked DaMya when she got, DaMya said she got t=23.79 and Ms. Mota said it was right.
What did Dearra do wrong?
Dearra brought down the t without getting the log of both sides. The only reason you can "bring down the t" is because of the property of logarithms that allows you to bring down an exponent. Dearra brought down the exponent without getting the log of both sides. She should have gotten t=log(5)/log(1.07)=23.79.
Solo said the function is f(t)=8(1.5)t, where t=20 and Monty said the the function is f(t)=8(1+1.5)t, where t=4.
Who is right and who is wrong? Why?Surprise! Both are wrong. While Solo got her function correct, he input is incorrect. Since the chunk of time is 5 years, and 20 years have passes, that means t=4. Meanwhile, Monty get his t correct, but got his function wrong. When something grows by a certain factor, in this case 1.5, we use the function format of f(t)=P(r)t rather than f(t)=p(1±r)t like Monty did.
A species of amoeba is known to double every three days. If the the known population of amoeba in a lake were 2.5 million on the morning of February 1st, on the morning of what date would the population of amoeba be expected to reach 80 million? You must use logarithms to solve and be prepared to justify your answer.
What is the morning of February 16th?