6.8 Exponential and Logarithmic Models

6.8 Exponential and Logarithmic Models In this section, we will study the following topics: Using exponential growth and decay functions to solve real-life problems Using logistic growth functions to solve real-life problems Using logarithmic functions to solve real-life problems

Five Common Mathematical Models Many business applications and natural phenomena can be modeled by exponential and logarithmic functions. In this section we will use exponential, logistic, and logarithmic models to solve some real-life applications.

An Exponential Growth Model • Another way of writing the exponential growth model is • If k > 0, the function models the size of a growing entity. • If k < 0, the function models the size of a decaying entity. • A0is the original amount, or size, of the entity at time t = 0. • A is the amount at time t. • k is a constant representing the growth rate.

An Exponential Growth Model • Example: • In 1970, the U.S. population was 203.3 million. By 2003, it had grown to 294 million. • Find the exponential growth function that models the U.S. population growth for 1970 through 2003. • According to this exponential model, in which year will the U.S. population reach 315 million?

An Exponential Growth Model(continued) Solution: Use the model . In this example, A represents ___________________________________________ A0 represents __________________________________________ t represents ___________________________________________ Since the population is increasing, k is a ______________ number. Substitute the value ________ for A0 in the growth model:

An Exponential Growth Model(continued) We are given that the population in 2003 was 294 million. 2003 is _________ years after 1970. Therefore, when t = ______, the value of A is ___________. *On the graph, this would be represented as the point ( , ) Substitute these values into the growth model to find the value of k, the growth rate:

An Exponential Growth Model(continued) Now we substitute the value of k into the growth model to obtain the exponential growth model for the U.S. population. It is: To find the year in which the population will reach 315 million, substitute 315 for A in the model we just found and solve for t.

Applications ofLogistic Growth Models Logistic growth functions are used to model situations where initially there is an increasing rate of growth followed by a decreasing rate of growth. Some common examples include the spread of a disease within a population and the growth of certain populations. In these cases, the functions have an upper bound which is equal to the maximum population capacity. This upper bound and the line y = 0 are the horizontal asymptotes of the graphs of logistic functions.

Horizontal asymptote y=c provides a limit to growth. Original amount at x=0 Graph of Logistic Model Domain: (-, ) Range: (0, c) Horizontal Asymptotes: y = 0, y = c

A Logistic Growth Model • Example: • Fruit flies are placed in a half-pint milk bottle with a banana (for food) and yeast plants (for food and to provide stimulus to lay eggs). Suppose that the fruit fly population P after t days is given by • What is the carrying capacity of the half-pint bottle? (That is, what is the upper limit of the population?) • How many fruit flies were initially placed in the half-pint bottle? • When will the population be 180?

A Logistic Growth Model(continued) • Solution: • We can use the TABLE feature on the graphing calculator table to find P(t) as t→. y = 230 y = 0 P approaches 230 as t gets increasingly larger. Notice, the LIMITING CAPACITY IS THE NUMERATOR OF THIS FUNCTION. It is also the upper horizontal asymptote of the graph of this function. (y = 0 is the lower asymptote). So, the carrying capacity of the bottle is _______ fruit flies.

A Logistic Growth Model(continued) • Solution: • To find the initial number of fruit flies in the bottle, we need to find P(0). • Algebraically: Graphically: Initially, there were ________ fruit flies in the half-pint bottle.

A Logistic Growth Model(continued) • Solution: • To find when the population will reach 180, set P(t)=180. • Algebraically: Graphically: It will take approx. _______ days for the pop. to reach 180 fruit flies.

Applications of Logarithmic Models • Many relations between variables are best modeled by a logarithmic function. • Some common examples include: • the relation between an earthquake’s magnitude and intensity on the Richter scale, • the relation between atmospheric pressure and height, • the relation between sound level (in decibels) and intensity, • as well as many economic models. • Refer to Example 6 in your text.

A Logarithmic Model Example: The loudness L, in bels (named after ?????), of a sound of intensity I is defined to be where I0 is the minimum intensity detectable by the human ear. The bell is a large unit, so a subunit, the decibel, is generally used. For L, in decibels, the formula is

A Logarithmic Model(cont) Find the loudness, in decibels, for each sound with the given intensity. Library Dishwasher Loud muffler

End of Sect. 6.8

6.8 Exponential and Logarithmic Models