File Name: exponential and logarithmic equations and inequalities writer.zip
We have already explored some basic applications of exponential and logarithmic functions. In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth, we may choose the exponential growth function:.
We may use the exponential growth function in applications involving doubling time , the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time.
In some applications, however, as we will see when we discuss the logistic equation, the logistic model sometimes fits the data better than the exponential model. On the other hand, if a quantity is falling rapidly toward zero, without ever reaching zero, then we should probably choose the exponential decay model.
We may use the exponential decay model when we are calculating half-life, or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes. In our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph. Exponential growth and decay often involve very large or very small numbers.
To describe these numbers, we often use orders of magnitude. The order of magnitude is the power of ten, when the number is expressed in scientific notation, with one digit to the left of the decimal. A population of bacteria doubles every hour. When an amount grows at a fixed percent per unit time, the growth is exponential. The formula is derived as follows. We now turn to exponential decay. One of the common terms associated with exponential decay, as stated above, is half-life , the length of time it takes an exponentially decaying quantity to decrease to half its original amount.
Every radioactive isotope has a half-life, and the process describing the exponential decay of an isotope is called radioactive decay. To find the half-life of a function describing exponential decay, solve the following equation:. This gives us the half-life formula. Find function gives the amount of carbon remaining as a function of time, measured in years. The formula for radioactive decay is important in radiocarbon dating , which is used to calculate the approximate date a plant or animal died.
Radiocarbon dating was discovered in by Willard Libby, who won a Nobel Prize for his discovery. It compares the difference between the ratio of two isotopes of carbon in an organic artifact or fossil to the ratio of those two isotopes in the air. It occurs in small quantities in the carbon dioxide in the air we breathe. As long as a plant or animal is alive, the ratio of the two isotopes of carbon in its body is close to the ratio in the atmosphere.
When it dies, the carbon in its body decays and is not replaced. By comparing the ratio of carbon to carbon in a decaying sample to the known ratio in the atmosphere, the date the plant or animal died can be approximated.
Out of necessity, we neglect here the many details that a scientist takes into consideration when doing carbon dating, and we only look at the basic formula. Given the percentage of carbon in an object, determine its age. To the nearest year, how old is the bone?
The instruments that measure the percentage of carbon are extremely sensitive and, as we mention above, a scientist will need to do much more work than we did in order to be satisfied.
For decaying quantities, we determined how long it took for half of a substance to decay. For growing quantities, we might want to find out how long it takes for a quantity to double. As we mentioned above, the time it takes for a quantity to double is called the doubling time. Give a function that describes this behavior. Growth has slowed to a doubling time of approximately three years. Find the new function that takes that longer doubling time into account.
Exponential decay can also be applied to temperature. On a graph of the temperature function, the leveling off will correspond to a horizontal asymptote at the temperature of the surrounding air. Unless the room temperature is zero, this will correspond to a vertical shift of the generic exponential decay function.
Exponential growth cannot continue forever. Exponential models, while they may be useful in the short term, tend to fall apart the longer they continue. Consider an aspiring writer who writes a single line on day one and plans to double the number of lines she writes each day for a month.
It is impractical, if not impossible, for anyone to write that much in such a short period of time. Eventually, an exponential model must begin to approach some limiting value, and then the growth is forced to slow. For this reason, it is often better to use a model with an upper bound instead of an exponential growth model, though the exponential growth model is still useful over a short term, before approaching the limiting value.
The graph increases from left to right, but the growth rate only increases until it reaches its point of maximum growth rate, at which point the rate of increase decreases.
An influenza epidemic spreads through a population rapidly, at a rate that depends on two factors: The more people who have the flu, the more rapidly it spreads, and also the more uninfected people there are, the more rapidly it spreads.
These two factors make the logistic model a good one to study the spread of communicable diseases. And, clearly, there is a maximum value for the number of people infected: the entire population. Estimate the number of people in this community who will have had this flu after ten days.
Predict how many people in this community will have had this flu after a long period of time has passed. Remember that, because we are dealing with a virus, we cannot predict with certainty the number of people infected. The model only approximates the number of people infected and will not give us exact or actual values. Now that we have discussed various mathematical models, we need to learn how to choose the appropriate model for the raw data we have.
Many factors influence the choice of a mathematical model, among which are experience, scientific laws, and patterns in the data itself. Not all data can be described by elementary functions. Sometimes, a function is chosen that approximates the data over a given interval. For instance, suppose data were gathered on the number of homes bought in the United States from the years to After plotting these data in a scatter plot, we notice that the shape of the data from the years to follow a logarithmic curve.
We could restrict the interval from to , apply regression analysis using a logarithmic model, and use it to predict the number of home buyers for the year Three kinds of functions that are often useful in mathematical models are linear functions, exponential functions, and logarithmic functions.
If the data lies on a straight line, or seems to lie approximately along a straight line, a linear model may be best. If the data is non-linear, we often consider an exponential or logarithmic model, though other models, such as quadratic models, may also be considered. In choosing between an exponential model and a logarithmic model, we look at the way the data curves. This is called the concavity. If we draw a line between two data points, and all or most of the data between those two points lies above that line, we say the curve is concave down.
We can think of it as a bowl that bends downward and therefore cannot hold water. If all or most of the data between those two points lies below the line, we say the curve is concave up. In this case, we can think of a bowl that bends upward and can therefore hold water. An exponential curve, whether rising or falling, whether representing growth or decay, is always concave up away from its horizontal asymptote.
A logarithmic curve is always concave away from its vertical asymptote. In the case of positive data, which is the most common case, an exponential curve is always concave up, and a logarithmic curve always concave down. A logistic curve changes concavity. It starts out concave up and then changes to concave down beyond a certain point, called a point of inflection.
After using the graph to help us choose a type of function to use as a model, we substitute points, and solve to find the parameters. We reduce round-off error by choosing points as far apart as possible. Find the model, and use a graph to check your choice. For the purpose of graphing, round the data to two significant digits. Clearly, the points do not lie on a straight line, so we reject a linear model. If we draw a line between any two of the points, most or all of the points between those two points lie above the line, so the graph is concave down, suggesting a logarithmic model.
Find the model. Access these online resources for additional instruction and practice with exponential and logarithmic models. Jay Abramson Arizona State University with contributing authors. Learning Objectives Model exponential growth and decay. Use logistic-growth models. Choose an appropriate model for data. Modeling Exponential Growth and Decay In real-world applications, we need to model the behavior of a function.
Solution When an amount grows at a fixed percent per unit time, the growth is exponential. Half-Life We now turn to exponential decay. Solution This formula is derived as follows. Radiocarbon Dating The formula for radioactive decay is important in radiocarbon dating , which is used to calculate the approximate date a plant or animal died.
Analysis The instruments that measure the percentage of carbon are extremely sensitive and, as we mention above, a scientist will need to do much more work than we did in order to be satisfied. Calculating Doubling Time For decaying quantities, we determined how long it took for half of a substance to decay. Substitute in the desired time to find the temperature or the desired temperature to find the time. Using Logistic Growth Models Exponential growth cannot continue forever.
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This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove. Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for x :. The one-to-one property does not help us in this instance.
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