# Exponential and Logarithmic Functions

## Exponential and Logarithmic Models


### Learning Objectives

1. Use exponential models in applications. (IA 10.2.3)
2. Use logarithmic models in applications. (IA 10.3.5)

### Objective 1: Use exponential models in applications. (IA 10.2.3)


### Using exponential models

Exponential functions model many situations. If you have a savings account, you have experienced the use of an exponential function. There are two formulas that are used to determine the balance in the account when interest is earned. If a principal, P, is invested at an interest rate, r, for t years, the new balance, A, will depend on how often the interest is compounded.

### Exponential Growth and Decay

Other topics that are modeled by exponential functions involve growth and decay. Both also use the formula  we used for the growth of money. For growth and decay, generally we use  as the original amount instead of calling it  the principal. We see that exponential growth has a positive rate of growth and exponential decay has a negative rate of growth.

### Practice Makes Perfect


### Objective 2: Use logarithmic models in applications. (IA 10.3.5)


### Decibel Level of Sound


There are many applications that are modeled by logarithmic equations. We will first look at the logarithmic equation that gives the decibel (dB) level of sound. Decibels range from 0, which is barely audible to 160, which can rupture an eardrum. The10-12 in the formula represents the intensity of sound that is barely audible.

The magnitude  of an earthquake is measured by a logarithmic scale called the Richter scale. The model is  where  is the intensity of the shock wave. This model provides a way to measure earthquake intensity.

### Practice Makes Perfect

Use logarithmic models in applications.
We have already explored some basic applications of exponential and logarithmic functions. In this section, we explore some important applications in more depth, including radioactive isotopes and Newton’s Law of Cooling.

### Modeling Exponential Growth and Decay

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:
where  is equal to the value at time zero,  is Euler’s constant, and  is a positive constant that determines the rate (percentage) of growth. 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. Again, we have the form  where  is the starting value, and  is Euler’s constant. Now  is a negative constant that determines the rate of decay. 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 graphs have a distinctive shape, as we can see in  and . It is important to remember that, although parts of each of the two graphs seem to lie on the x-axis, they are really a tiny distance above the x-axis.
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. For example, the distance to the nearest star, Proxima Centauri, measured in kilometers, is 40,113,497,200,000 kilometers. Expressed in scientific notation, this is  So, we could describe this number as having order of magnitude  

### Half-Life

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:
We find that the half-life depends only on the constant  and not on the starting quantity  
The formula is derived as follows

Since  the time, is positive,  must, as expected, be negative. This gives us the half-life formula

### 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. Radiocarbon dating was discovered in 1949 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 is believed to be accurate to within about 1% error for plants or animals that died within the last 60,000 years.
Carbon-14 is a radioactive isotope of carbon that has a half-life of 5,730 years. It occurs in small quantities in the carbon dioxide in the air we breathe. Most of the carbon on Earth is carbon-12, which has an atomic weight of 12 and is not radioactive. Scientists have determined the ratio of carbon-14 to carbon-12 in the air for the last 60,000 years, using tree rings and other organic samples of known dates—although the ratio has changed slightly over the centuries.
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-14 in its body decays and is not replaced. By comparing the ratio of carbon-14 to carbon-12 in a decaying sample to the known ratio in the atmosphere, the date the plant or animal died can be approximated.
Since the half-life of carbon-14 is 5,730 years, the formula for the amount of carbon-14 remaining after  years is

where


This formula is derived as follows:
To find the age of an object, we solve this equation for  
Out of necessity, we neglect here the many details that a scientist takes into consideration when doing carbon-14 dating, and we only look at the basic formula. The ratio of carbon-14 to carbon-12 in the atmosphere is approximately 0.0000000001%. Let  be the ratio of carbon-14 to carbon-12 in the organic artifact or fossil to be dated, determined by a method called liquid scintillation. From the equation  we know the ratio of the percentage of carbon-14 in the object we are dating to the initial amount of carbon-14 in the object when it was formed is  We solve this equation for  to get

### Calculating Doubling Time

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.
Given the basic exponential growth equation 
 doubling time can be found by solving for when the original quantity has doubled, that is, by solving 
 
The formula is derived as follows:
Thus the doubling time is

### Using Newton’s Law of Cooling

Exponential decay can also be applied to temperature. When a hot object is left in surrounding air that is at a lower temperature, the object’s temperature will decrease exponentially, leveling off as it approaches the surrounding air 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. This translation leads to Newton’s Law of Cooling, the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature
This formula is derived as follows:

### Using Logistic Growth Models

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. By the end of the month, she must write over 17 billion lines, or one-half-billion pages. 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 logistic growth model is approximately exponential at first, but it has a reduced rate of growth as the output approaches the model’s upper bound, called the carrying capacity. For constants 
 and 
 the logistic growth of a population over time 
 is represented by the model
The graph in  shows how the growth rate changes over time. 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.

### Choosing an Appropriate Model for Data

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 1960 to 2013. After plotting these data in a scatter plot, we notice that the shape of the data from the years 2000 to 2013 follow a logarithmic curve. We could restrict the interval from 2000 to 2010, apply regression analysis using a logarithmic model, and use it to predict the number of home buyers for the year 2015.
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.

### Expressing an Exponential Model in Base  

While powers and logarithms of any base can be used in modeling, the two most common bases are 
 and 
 In science and mathematics, the base 
 is often preferred. We can use laws of exponents and laws of logarithms to change any base to base 
 

### Key Equations


### Key Concepts

1. The basic exponential function is 
 If 
 we have exponential growth; if 
 we have exponential decay.
2. We can also write this formula in terms of continuous growth as 
 where 
 is the starting value. If 
 is positive, then we have exponential growth when 
 and exponential decay when 
 See .
3. In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay. See .
4. We can find the age, 
 of an organic artifact by measuring the amount, 
 of carbon-14 remaining in the artifact and using the formula 
 to solve for 
 See .
5. Given a substance’s doubling time or half-time, we can find a function that represents its exponential growth or decay. See .
6. We can use Newton’s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time. See .
7. We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors. See .
8. We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data. See .
9. Any exponential function with the form 
 can be rewritten as an equivalent exponential function with the form 
 where 
 See .

### Section Exercises


### Verbal


### Numeric

For the following exercises, use the logistic growth model 
 

### Technology

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.
For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in  
 years is modeled by the equation 
 

### Extensions


### Real-World Applications

For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.
For the following exercises, use this scenario: A tumor is injected with 
 grams of Iodine-125, which has a decay rate of 
  
 per day.
For the following exercises, use this scenario: A biologist recorded a count of 
 bacteria present in a culture after 5 minutes and 1000 bacteria present after 20 minutes.
For the following exercises, use this scenario: A pot of warm soup with an internal temperature of 
 Fahrenheit was taken off the stove to cool in a 
 room. After fifteen minutes, the internal temperature of the soup was 
 
For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of 
 and is allowed to cool in a 
 room. After half an hour, the internal temperature of the turkey is 
 
For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.
For the following exercises, use this scenario: The equation 
 models the number of people in a town who have heard a rumor after t days.
For the following exercise, choose the correct answer choice.