Write an Exponential Function Given a Rate and an Initial Value

Exponential functions can model the rate of change of many situations, including population growth, radioactive decay, bacterial growth, compound interest, and much more. Follow these steps to write an exponential equation if you know the rate at which the function is growing or decaying, and the initial value of the group.

Steps

Using the Rate as the Base

  1. Consider an example. Suppose a bank account is started with a $1,000 deposit and the interest rate is 3% compounded annually. Find an exponential equation modeling this function.
  2. Know the basic form. The form for an exponential equation is f(t)=P0(1+r)t/h where P0 is the initial value, t is the time variable, r is the rate and h is the number needed to ensure the units of t match up with the rate.
  3. Plug in the initial value for P0 and the rate for r. You will have f(t)=1,000(1.03)t/h.
  4. Find h. Think about your equation. Every year, the money increases by 3%, so every 12 months the money increases by 3%. Since you need to give t in months, you have to divide t by 12, so h=12. Your equation is f(t)=1,000(1.03)t/12. If the units are the same for the rate and the t increments, h is always 1.

Using "e" as the base

  1. Understand what e is. When you use the value e as the base, you are using the "natural base." Using the natural base allows you to pull the continuous growth rate directly from the equation.
  2. Consider an example. Suppose a 500 gram sample of an isotope of Carbon has a half life of 50 years (the half life is the amount of time for the material to decay by 50%).
  3. Know the basic form. The form for an exponential equation is f(t)=aekt where a is the initial value, e is the base, k is the continuous growth rate, and t is the time variable.
  4. Plug in the initial value. The only value you are given that you need in the equation is the initial growth rate. So, plug it in for a to get f(t)=500ekt
  5. Find the continuous growth rate. The continuous growth rate is how fast the graph is changing at a particular instant. You know that in 50 years, the sample will decay to 250 grams. That can be considered a point on the graph that you can plug in. So t is 50. Plug it in to get f(50)=500e50k. You also know that f(50)=250, so substitute 250 for f(50) on the left hand side to obtain the exponential equation 250=500e50k. Now to solve the equation, first divide both sides by 500 to get: 1/2=e50k. Then take the natural logarithm of both sides to get: ln(1/2)=ln(e50k. Use the properties of logarithms to take the exponent out of the argument of the natural log and multiply it by the log. This results in ln(1/2)=50k(ln(e)). Recall that ln is the same thing as loge and that the properties of logarithms say that if the base and the argument of the logarithm are the same, the value is 1. Therefore ln(e)=1. So the equation simplifies to ln(1/2)=50k, and if you divide by 50, you learn that k=(ln(1/2))/50. Use your calculator to find the decimal approximation of k to be approximately -.01386. Notice that this value is negative. If the continuous growth rate is negative, you have exponential decay, if it is positive, you have exponential growth.
  6. Plug in the k value. Your equation is 500e-.01386t.

Tips

  • You might want to store your k value in your calculator so you can calculate your values more exactly than with a decimal approximation. X is an easily accessible variable to use since you don't need to press "alpha" to get to it, but if you want to graph the equation, be sure to use a variable designated as a constant or you'll put in extra variables.
  • You will quickly learn when to use each method. Usually, problems are easier using the first method, but there are times when you know using the natural base will make your calculations easier later.

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