Exponential Growth and Decay

Key Questions

• How to find an equation of exponential decay?

  An equation of exponential decay typically takes the form `A(t) = A(0)e^(kt)` where `k<0`, though sometimes these will be written as `A(0)e^(-kt)` and have `k>0`. Either form is acceptable, though some argue that the first form is more accurate, so that is the form that shall be used here. Note that there are several possible sets of information you may be given.

  As your question deals specifically with finding the equation (as opposed to the value the function takes at some time `t=m`) you will typically be asked to find either `k`, or more rarely `A(0)` (also written sometimes as `A_0`). In a case where one is being asked to find `k`, typically one has been given the value for `A_0` (the initial value of the function at time `t=0`) and `A(m)` (the value of `A(t)` at time `t=m`. From this, we can find the value of `k` and complete our function as follows:

  `A(m) = A_0 e^(km)`
  `(A(m))/A_0 = e^(km)`
  `ln((A(m))/A_0) = km` (remember that `ln(e^x) = x`)
  `ln(A(m)) - ln(A_0) = km`
  `[ln(A(m))-ln(A_0)]/m = k`

  Then this value of `k` can be substituted in, and since we know `A_0`, we can complete our function of `A(t) = A_0e^(kt)` (note that `t` is still our independent variable, `A(t)` our dependent variable, and `e` a constant approx. equal to 2.718282, defined such that `ln(e) = 1`

  Alternately, if one is tasked with finding `A(0)`, one has likely already been given `k` and the value `A(m)` of the function at some time `t=m`. In this case, our process is far easier.

  `A(m) = A_0e^(km)`

  Since we have values for `A(m), k,` and `m` (which is our value for `t` at this point, recall), and since an exponential function cannot equal 0 unless the base equals 0, we can simply perform division to obtain:

  `(A(m))/(e^(km)) = A_0`

  Psykolord1989 . · 1 · Oct 31 2014
• What is exponential growth?

  Exponential growth is basically growth that begins at a slow rate, but then gets faster as it goes. On a graph it looks like this:

  ![www.quia.com]()

  Exponential growth can be modelled using the following equation:

  `y = ab^(x-h)+k`

  This video helps explain how exponential functions work:

  Intro to Exponential Functions

  Exponential growth is also a concept related to population growth that you will see in ecology. This video helps explain how it works:

  Introduction to Exponential Growth in Ecology

  Darshan Senthil · · Sep 11 2014

• How do I find an exponential growth function in terms of `t`?
• What is exponential growth?
• How do I find the multiplier for a rate of exponential decay?
• How is exponential decay related to a half-life?
• How to find an equation of exponential decay?
• What happens when something grows exponentially?
• How long does it take the culture to double its mass if a bacterial culture which is growing exponentially increases from 5 g to 7 g in 11 hours?
• Joe Smith invest his inheritance of $50,000 in an account paying 6.5% interest. If interest is compounded continuously, how long will it take for the account to be $200,000?
• How do you write an exponential equation that passes through (1, 1.5), (-1, 6)?
• How do you write an exponential equation that passes through (0,3) and (2,6)?
• How do you write an exponential equation that passes through (-1, .5) and (2, 10)?
• How do you write an exponential equation that passes through (0, -2) and (2, -50)?
• Is `y = -3^x` an exponential function?
• Question #70b92
• A certain species of fish has been declining in population by 7% per year. That means its rate of decay is 0.93. What is the fish population in year 4 if the initial population was 500,000?
• What is the balance if $20,000 is invested at an annual rate of 7.8 percent for 5 years, compounded continuously. How long will it take for the original amount to double in size?
• Question #44bee
• With 300 being the y intercepts and an additional point being (3,450) How do you find a formula in the format of `ab^x`?
• How do you find the exponential function f(x)= a^x whose graph goes through the point (3, 1/125)?
• A bacteria culture initially contains 1500 bacteria and doubles every half hour. The formula for the population is p(t)= 1500e^kt for some constant k, how do you find the size of the population after 60 minutes?
• How do you determine the formula for the graph of an exponential function given (0,3) and (2,6)?
• How do you determine the formula for the graph of an exponential function given (0,2) and (3,1)?
• How do you state two equations for an exponential function that passes through (0,12) and shows growth?
• How do you state an equation for an exponential function that passes through (0,7) and shows decay?
• How would you find P and A of the equation f(x)=PA^x given f(2)=5400 and f(6)= 345000?
• How do you write an exponential function to model each situation & solve given the population of Warren Michigan was 145,600 in 1990. If the population decreases by 1.2% each year, predict the population in 1999?
• How do you write an exponential function to model each situation & solve given whole milk consumption in the U.S. has decreased by 4% annually since 1985. Each person consumed 13.6 gallons of whole milk in 1985. Predict whole milk consumption in 2000?
• What is the domain of f(x) = (-2)^x?
• Why is x^-3 not a Exponential Function?
• Which is an exponential function f(x)=x^3 or g(x)=3^x?
• How do you write an exponential function of the form y = a b^x the graph of which passes through (-1, 4/5) and (2, 100)?
• How do you find formula for the exponential function in the form of f(x)= Ca^x given f(0)=3 and f(1)=15?
• The half life of the radioactive element Strontium-90 is 37 years. In 1950, 15 kilograms of this element released accidentally. How do you determine the formula which shows the mass remaining after t years?
• A lab has 40 kg of radioactive uranium this substance has a half-life of 8 minutes and decays exponentially. How do you find the decay constant k?
• How do you find the y-intercept of `y=10(4^x)`?
• If 2^x = 3, what does 4^(-x) equal?
• If `5^(-x) = 3`, what does `5^(3x)` equal?
• How do you find an exponential function of the form f(x)=ab^x that has the given y-intercept (0,5); passing through P=(2, 5/9)?
• How do you find an exponential function of the form f(x)=ab^x that has the given y-intercept (0,2); passing through P=(-1, 2/5)?
• How do you find formulas for the exponential functions satisfying the given conditions h(0)=3 and h(1)= 15?
• How do you find formulas for the exponential functions satisfying the given conditions f(3)=-3/8 and f(-2)=-12?
• How do you find formulas for the exponential functions satisfying the given conditions g(1/2)=4 and g(1/4)=2sqrt2?
• A serving of a popular cola soft drink has 46 mg of caffeine. Suppose the half-life for caffeine remaining in he body of a typical adult is 6 h. How do you write an exponential decay function that gives the amount of caffeine in the body after t hours?
• How do you find the exponential function f(x) = Ca^x whose graph has the given points. (2,0) and (2,2/9)?
• How do you find the exponential function f(x) = Ca^x whose graph has the given points (0,2) and (3,54)?
• Radium-221 has a half-life of 30 seconds. How long will it take for 94% of a sample to decay?
• If 250 mg of a radioactive element decays to 220 mg in 12 hours, how do you find the half-life of the element?
• The half-life of radium is 1590 year. If 10 grams is present now, how much will be present in 50 years?
• Use the exponential decay model, A=Aoe^(kt), to solve this exercise. The half-life of polonium-210 is 140 days. How long will it take for a sample of this substance to decay to 20% of its original amount?
• A population P is initially 1000. How do you find an exponential model (growth or decay) for the population after t years if the population P is increases by 200% every 6 years?
• The value of an industrial machine has a decay factor of 0.75 per year. After six years, the machine is worth 7500. What was the original value of the machine?
• In the year 2000, you invested money in a money market account. The value of your investment t years after 2000 is given by the exponential growth model A = 1900e^(0.058t). By what percentage is the account increasing each year?
• A population P is initially 3000. How do you find an exponential model (growth or decay) for the population after t years if the population P is multiplied by 0.65 every month?
• The half-life of cobalt 60 is 5 years. How do you obtain an exponential decay model for cobalt 60 in the form Q(t) = Q0e^−kt?
• How do you find the associated exponential decay model given Q = 100 when t = 0; Half-life = 6?
• Population in 1990 was 643 million. By 2000, it had grown to 813 million. HOw do you find the exponential growth model?
• The half-life of iodine-131 is 7.2 days. How long will it take for a sample of this substance to decay to 30% of its original amount?
• A used car has a value of $15,250 when it is purchased in 2012. The value of the car depreciates at a rate of 7.5% per year. How do you write an exponential function that models the value of the car, y, over x years?
• The function `f(x) = 200 × (1.098)^x` represents a village's population while it is growing at the rate of 9.8% per year. How do you create a table to show the village's population at 0, 2, 4, 6, 8, and 10 years from now?
• A sample of 2100 bacteria selected from this population reached the size of 2169 bacteria in one and a half hours. How do you find the hourly growth rate parameter?
• You Buy a Commemorative coin for $110. Each Year (t) the value (v) of the coin increases by 4%. Write an exponential model for this situation. Calculate the value of the coin after 3 years. When will the value of the coin will be $150?
• A bacteria culture starts with 2,000 bacteria and doubles in size every 3 hours. How do you find an exponential model for the size of the culture as a function of time t in hours?
• How do you find the exponential model `y=ae^(bx)` that goes through the points (4,256) (3,64)?
• A bacteria culture starts with 1,500 bacteria and doubles in size every 2 hours. How do you find an exponential model for the size of the culture as a function of time t in hours?
• How do you find the exponential model `y=ae^(bx)` that goes through the points (1, 1.2) and (4, 0.0768)?
• A bacterial culture starts with 2,000 bacteria and doubles in size every 6 hours. How do you find an exponential model for the size of the culture as a function of time t in hours?
• How do you find the exponential model `y=ae^(bx)` that goes through the points (0,7) (5, 32)?
• How do you find the exponential model `y=ae^(bx)` that goes through the points (2,8) (6, 128)?
• How do you find the exponential model `y=ae^(bx)` that goes through the points (-3,2)(1,164)?
• How do you find the exponential model `y=ae^(bx)` that goes through the points (-1,125) (2,5)?
• How do you find the exponential model `y=ae^(bx)` that goes through the points (-5, 243/2) and (-1, 3/2)?
• How do you find the exponential model `y=ae^(bx)` that goes through the points (5,3200) and (11,6800)?
• The decay rate of a certain chemical is 9.8% per year what is the half life?
• How do you find the associated exponential decay or growth model: Q = 1,500 when t = 0; half-life = 1?
• The value of a new car bought for $20,000 decreased by 10% per year. How do you write an exponential decay model for the value of the car and use the model to estimate the value after one year?
• The population of the world in 1987 was 5 billion and the relative growth rate was estimated at 2 percent per year. Assuming that the world population follows an exponential growth model, find the projected world population in 1995?
• Question #ae84e
• What is an example of a value of b for which `y=b^x` represents an exponential decay?
• How do you graph `y=5^x`?
• How do you graph `y=2(5)^x`?
• How do you graph `y=3(4)^x` and state the domain and range?
• How do you graph `y=(0.5)^x` and state the domain and range?
• How do you graph `y=0.3(5)^x` and state the domain and range?
• How do you write an exponential function whose graph passes through the points (0,3) and (-1,6)?
• How do you write an exponential function whose graph passes through the points (0,-18) and (-2,-2)?
• How do you simplify `2^sqrt7*2^sqrt7`?
• How do you simplify `(a^pi)^4`?
• How do you simplify `81^sqrt2 div 3^sqrt2`?
• How do you graph of `y=2(3)^x` and state the domain and range?
• How do you graph of `y=5(2)^x` and state the domain and range?
• How do you graph of `y=-(1/5)^x` and state the domain and range?
• How do you determine if `y=2(4)^x` is an exponential growth or decay?
• How do you determine if `y=0.4(1/3)^x` is an exponential growth or decay?
• How do you determine if `y=3(5/2)^x` is an exponential growth or decay?
• How do you determine if `y=30^-x` is an exponential growth or decay?
• How do you determine if `y=0.2(5)^-x` is an exponential growth or decay?
• How do you write an exponential function whose graph passes through (0,3) and (1,15)?
• How do you write an exponential function whose graph passes through (0,7) and (2,63)?
• How do you write an exponential function whose graph passes through (0,-5) and (-3, -135)?
• How do you write an exponential function whose graph passes through (0,0.2) and (4, 51.2)?
• How do you write an exponential function whose graph passes through (0,-0.3) and (5,-9.6)?
• How do you simplify `(5^sqrt2)^sqrt8`?
• How do you simplify `(x^sqrt5)^sqrt3`?
• How do you simplify `7^sqrt2*7^(3sqrt2)`?
• How do you simplify `y^(3sqrt3)divy^sqrt3`?
• How do you simplify `n^2*n^pi`?
• How do you simplify `64^pidiv2^pi`?
• How do you graph `y=3^x` and `y=3^(x+1)` and how do the graphs compare?
• How do you graph `y=(1/5)^x` and `y=(1/5)^(x-2)` and how do the graphs compare?
• How do you graph `y=(1/4)^x` and `y=(1/4)^x-1` and how do the graphs compare?
• What are the effect of changing the values of h and k in the equation `y=2^(x-h)+k`?
• How do you simplify `(b^sqrt6)^sqrt24`?
• A country's population in 1994 was 195 million. In 2002 it was 199 million. How do you estimate the population in 2016 using the exponential growth formula `P=Ae^(kt)`?
• How do you find the equation of the exponential function `y=a(b)^x` which goes through the points(2, 18) and (6,91.125). What is the intercepts of this function?
• The initial population of a large city is `20` million. Its population increases by 2% each year. It's 2012 today. How many people will live in the city at the end of 2016?
• The exponential function `f(x)= 15,000(1.02)^x` models the amount of money in a savings account over a period of time. What does the value 15,000 represent?
• How do you find the annual percent increase or decrease that `y =0.35(2.3)^x` models?
• A biologist studying a population of geckos for several years found that the population can be modeled as where `P` is the population, and `t` is in the time in years. `P= 230( 1.04)^t` . What is the growth rate of this population?
• How do you evaluate the function `f(x)=2.3^x` at the value of `x=3/2`?
• How do you evaluate the function `f(x)=5^x` at the value of `x=-pi`?
• How do you evaluate the function `g(x)=5000(2^x)` at the value of `x=-1.5`?
• How do you evaluate the function `h(x)=e^-x` at the value of `x=3/4`?
• How do you evaluate the function `f(x)=e^x` at the value of `x=3.2`?
• How do you write the exponential functions goes through the points (1, 6) and (2, 12)?
• How do you graph and determine if `y=(4/3)^x` is a growth or decay?
• Question #553c3
• How do you tell whether `f(x)=(5/6)^x` is an exponential growth or decay?
• How do you tell whether `f(x)=(6/5)^x` is an exponential growth or decay?
• How do you tell whether `f(x)=(5/6)^-x` is an exponential growth or decay?
• How do you tell whether `f(x)=(6/5)^-x` is an exponential growth or decay?
• How do you tell whether `f(x)=-(6/5)^-x` is an exponential growth or decay?
• How do you graph, find any intercepts, domain and range of `f(x)=(1/4)^(x+3)-2`?
• How do you graph, find any intercepts, domain and range of `f(x)=4(1/2)^(x-1)-3`?
• How do you graph, find any intercepts, domain and range of `f(x)=(1/4)^(x+4)-3`?
• How do you graph, find any intercepts, domain and range of `f(x)=-(3/4)^(x+2)+5`?
• An exponential function passes through the points `(-3, 8)` and `(0, 1)`. What is the equation of the function?
• Question #2720a
• A population of rabbits follows the law of uninhibited growth. There are 5 rabbits initially and after 4 months there are 35. a.) How many rabbits will there be in one year? b.) How long will it take for the initial population to triple?
• When a living organism dies, the amount of Carbon-14 in its system begins to decrease exponentially. Given that the half-life of Carbon-14 is 5,730 years, answer the following: a.) What is the exact value of the rate of decay for Carbon-14?
• Question #e84e3
• Question #16386
• Question #72ac5
• The number of mold spores in a petri dish increases by a factor of 10 every week. If there are initially 40 spores in the dish, how long will it take for there to be 2000 spores?
• The value of a machine depreciate each year by 5% of its value at the beginning of that year. If its value when new on first 1968 was 10250,what was its value in January 1978?