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

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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.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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