Growth/Decay
What is the growth/decay factor in the exponential function below?
f(x)=4⋅1.5^x
1.5; growth
Consider the function
f(x) = 2^x
Does the graph of this function increase or decrease as x increases? (Is this growth or decay?)
Increases; the 2 implies this is exponential growth
Find the inverse of the function.
f(x) = 6^x
f^-1(x) = log_6(x)
Convert the exponential function to logarithmic:
2^6 = 64
log_2(64) = 6
A certain radioactive substance decays at a rate of 8% per year. Initially, there are 200 grams of the substance.
a) Write an equation that models the amount of the substance A(t) remaining after t years.
b) How much of the substance will remain after 5 years?
A(t) = 200(0.92)^t
A(5) = 131.82 grams
What is the horizontal asymptote of the function?
f(x)=3^x−4
y = -4
Find the inverse of the function:
f(x) = log_3(x)
f^-1(x) = 3^x
Simplify:
log_2(16)
log_2(2^4) = 4
A town’s population was 5,000 in the year 2020, and it has been growing at an annual rate of 3%.
a) Write an equation that models the population P(t) after t years.
b) Predict the population in the year 2030.
P(t) = 5000(1.03)^t
P(10) = 6719 people
Find the y-intercept of the exponential function:
f(x) = 5⋅(3)^(−x)
f(0) = 5*(3)^(-0)
= 5*1 = 5
(0,5)
Find the inverse of the function:
f(x) = 5^x−3
f ^(−1)(x) = log _5(x+3)
Solve for x:
log_9(3x+2) = log_9(5)
3x+2 = 5
3x = 3
x = 1
Suppose you invest $1,000 in an account that offers an annual interest rate of 5%, compounded quarterly.
a) Write an equation that models the amount of money A(t) in the account after t years.
b) How much money will be in the account after 6 years?
A(6) = 1000(1+(0.05/4))^(4t)
A(6) =1000(1+(0.05/4))^24
= $1347.35
A town has a measured population of 25,000 people in the year 2020. This town has a growth rate of 4% every year.
If we were to find a function to model this growth and plot it on a graph, what would the y-intercept actually represent?
The number of people in the town in 2020.
Find the inverse of the function:
f(x)=2ln(x+1)
f^(-1) = e^(x/2)-1
Simplify:
2log(x)+ 1/2log(y) − log(z)
log((x^2y^(1/2))/z)
The fictional substance Eridium has a half-life of just 5 hours when in contact with oxygen.
If 2,000 pounds of it are spilled in a town, how long would it take for there to be just 125 pounds left?
A(t) = 2000(0.5)^t
t = 4
4*5 = 20 hours
The function below models the decay of a substance in grams, where t represents time in years.
f(x) = 100⋅(0.9)^t
Determine after how many years the amount of substance will be less than 20 grams.
20 = 100*(0.9)^t
t = 16 years
Find the inverse of the function:
f(x) = 12⋅3^(x−4)+7
f ^(−1)(x) = log_3((x-7)/12)+4
Expand using product, quotient, and power rules:
log((xy)/z^2)
log(x) + log(y) - 2log(z)