# 4.7 Exponential and logarithmic models  (Page 6/16)

 Page 6 / 16

Using the model in [link] , estimate the number of cases of flu on day 15.

895 cases on day 15

## 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.

## Choosing a mathematical model

Does a linear, exponential, logarithmic, or logistic model best fit the values listed in [link] ? Find the model, and use a graph to check your choice.

 $x$ 1 2 3 4 5 6 7 8 9 $y$ 0 1.386 2.197 2.773 3.219 3.584 3.892 4.159 4.394

First, plot the data on a graph as in [link] . 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. We can try $\text{\hspace{0.17em}}y=a\mathrm{ln}\left(bx\right).\text{\hspace{0.17em}}$ Plugging in the first point, $\text{\hspace{0.17em}}\left(\text{1,0}\right)\text{,}\text{\hspace{0.17em}}$ gives $\text{\hspace{0.17em}}0=a\mathrm{ln}b.\text{\hspace{0.17em}}$ We reject the case that $\text{\hspace{0.17em}}a=0\text{\hspace{0.17em}}$ (if it were, all outputs would be 0), so we know $\text{\hspace{0.17em}}\mathrm{ln}\left(b\right)=0.\text{\hspace{0.17em}}$ Thus $\text{\hspace{0.17em}}b=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=a\mathrm{ln}\left(\text{x}\right).\text{\hspace{0.17em}}$ Next we can use the point $\text{\hspace{0.17em}}\left(\text{9,4}\text{.394}\right)\text{\hspace{0.17em}}$ to solve for $\text{\hspace{0.17em}}a:$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=a\mathrm{ln}\left(x\right)\hfill \\ 4.394=a\mathrm{ln}\left(9\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}a=\frac{4.394}{\mathrm{ln}\left(9\right)}\hfill \end{array}$

Because $\text{\hspace{0.17em}}a=\frac{4.394}{\mathrm{ln}\left(9\right)}\approx 2,$ an appropriate model for the data is $\text{\hspace{0.17em}}y=2\mathrm{ln}\left(x\right).$

To check the accuracy of the model, we graph the function together with the given points as in [link] .

We can conclude that the model is a good fit to the data.

Compare [link] to the graph of $\text{\hspace{0.17em}}y=\mathrm{ln}\left({x}^{2}\right)\text{\hspace{0.17em}}$ shown in [link] .

The graphs appear to be identical when $\text{\hspace{0.17em}}x>0.\text{\hspace{0.17em}}$ A quick check confirms this conclusion: $\text{\hspace{0.17em}}y=\mathrm{ln}\left({x}^{2}\right)=2\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}x>0.$

However, if $\text{\hspace{0.17em}}x<0,$ the graph of $\text{\hspace{0.17em}}y=\mathrm{ln}\left({x}^{2}\right)\text{\hspace{0.17em}}$ includes a “extra” branch, as shown in [link] . This occurs because, while $\text{\hspace{0.17em}}y=2\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ cannot have negative values in the domain (as such values would force the argument to be negative), the function $\text{\hspace{0.17em}}y=\mathrm{ln}\left({x}^{2}\right)\text{\hspace{0.17em}}$ can have negative domain values.

#### Questions & Answers

Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
how to understand calculus?
Jenica Reply

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