Exponentials Basics
State the initial value, whether it's growing or decaying, and by how much given the following equation:
y=4500(0.85)^x
4500, decaying by 15%
Write an exponential function in the form y=abx that goes through the points (0,18) and (3,144).
y=18(2)x
Write the exponential equation as a logarithmic equation.
9^-2=1/81
log_9(1/81)=-2
Round to the nearest hundredth.
62*10^x=40
x = -0.19
Describe a possible scenario for the following equation:
y=3500(1.14)^x
A population of bats starts at 3500 and grows at a rate of 14% per year.
A town has a population of 17000 and grows at 3% every year. To the nearest hundredth, how long will it be until the population will reach 23200?
23200=17000(1.03)x
x=10.52 years
State the initial value, whether it's growing or decaying, and by how much given the following equation:
y=10(1.031)^x
10, growth by 3.1%
Write an exponential function in the form y=abx that goes through the points (0,11) and (2,891).
y=11(9)x
Write the logarithmic equation as an exponential equation.
log_6(216)=3
6^3=216
Solve for x, round to the nearest hundredth.
756*3^(2x)=63
-1.13
Describe a possible scenario for the following equation:
f(x)=25900(0.93)^x
The car's value starts at $25,900 and depreciates by 7% each year.
A new car is purchased for 18500 dollars. The value of the car depreciates at 8% per year. To the nearest hundredth of a year, how long will it be until the value of the car is 9100 dollars?
9100=18500(0.92)x
x=8.51 years
A town has a population of 16,000 and grows at 3% every year. What will the population be after 7 years, to the nearest whole number?
16000(1.03)7=19678 people
Write an exponential function in the form y=abx that goes through the points (0,8200) and (4,9800).
y=8200(1.046)x
What is the value of
log_7(343)
3
Round to the nearest hundredth.
640e^(0.067t)=940
t = 5.74
Describe a possible scenario for the following equation:
g(x)=4500e^(0.08x)
An account starts at $4500 and earns interest at a rate of 8% compounded continuously.
A person invested $110 in an account growing at a rate allowing the money to double every 12 years. How long, to the nearest hundredth of a year would it take for the value of the account to reach $140?
140=110(2)^(t/12)
t = 4.18 years
An element with mass 350 grams decays by 26% per minute. How much of the element is remaining after 15 minutes? Round to the nearest hundredth.
350(0.74)15=3.82 grams
The population of a now endangered animal on an island is decaying exponentially. In the year 2009, the population was 2600, and by 2012 the it was only 1030. Write an equation to represent the number of animals, A, over time, t, in years.
y=2600(0.734)x
What is the value of
lne^2
2
Round to the nearest hundredth.
86*10^(x/4)=89
x = 0.06
Describe a possible scenario for the following equation:
g(t)=350000e^(0.1t)
The population of a city starts at 350,000 people and is continuously growing at a rate of 10% per year.
The metro are population of Dubai was 3,008,000 in 2023. Assuming the continuous growth rate stays at 1.42%, how long until the population reaches 3,250,000? Round to the nearest hundredth.
3,008,000e^(0.0142*x)=3250000
x = 5.45 years
The temperature of a cup of hot chocolate straight out of the microwave has a temperature of 165oF. It begins to cool at a rate of 4.8% per minute. What temperature will the hot chocolate be after 5 minutes? Round to the nearest hundredth.
165(0.952)5=129.02oF
Zachary purchased a new car in 2004 for $19,500. The value of the car has been depreciating exponentially at a constate percentage rate. If the value of the car was $13,300 in the year 2008, write an equation for the value of the car, V, in terms of t years.
V=19500(0.909)x
What is the value of
log_2(1/16)
-4
Round to the nearest hundredth.
log_4(3x+1)=3
x = 21
Describe a possible scenario for the following equation:
f(t)=103e^(-0.05t)
The temperature of bath water starts at 103oF and continuously cools down at a rate of 5% per minute.
Addison invested $440 in an account paying an interest rate of 2% compounded continuously. Assuming no deposits or withdrawals are made, how long would it take, to the nearest hundredth of a year, for the value of the account to reach $610?
610=440e^(0.02t)
t = 16.33 years