# 6.8 Fitting exponential models to data  (Page 4/12)

 Page 4 / 12

## Using logarithmic regression to fit a model to data

Due to advances in medicine and higher standards of living, life expectancy has been increasing in most developed countries since the beginning of the 20th century.

[link] shows the average life expectancies, in years, of Americans from 1900–2010 Source: Center for Disease Control and Prevention, 2013 .

 Year 1900 1910 1920 1930 1940 1950 Life Expectancy(Years) 47.3 50 54.1 59.7 62.9 68.2 Year 1960 1970 1980 1990 2000 2010 Life Expectancy(Years) 69.7 70.8 73.7 75.4 76.8 78.7
1. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represent time in decades starting with $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ for the year 1900, $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ for the year 1910, and so on. Let $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ represent the corresponding life expectancy. Use logarithmic regression to fit a model to these data.
2. Use the model to predict the average American life expectancy for the year 2030.
1. Using the STAT then EDIT menu on a graphing utility, list the years using values 1–12 in L1 and the corresponding life expectancy in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logarithmic pattern as shown in [link] :

Use the “LnReg” command from the STAT then CALC menu to obtain the logarithmic model,

$y=42.52722583+13.85752327\mathrm{ln}\left(x\right)$

Next, graph the model in the same window as the scatterplot to verify it is a good fit as shown in [link] :

2. To predict the life expectancy of an American in the year 2030, substitute $\text{\hspace{0.17em}}x=14\text{\hspace{0.17em}}$ for the in the model and solve for $\text{\hspace{0.17em}}y:$

If life expectancy continues to increase at this pace, the average life expectancy of an American will be 79.1 by the year 2030.

Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. [link] shows the number of games sold, in thousands, from the years 2000–2010.

 Year 2000 2001 2002 2003 2004 2005 Number Sold (thousands) 142 149 154 155 159 161 Year 2006 2007 2008 2009 2010 - Number Sold (thousands) 163 164 164 166 167 -
1. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represent time in years starting with $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ for the year 2000. Let $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ represent the number of games sold in thousands. Use logarithmic regression to fit a model to these data.
2. If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand.
1. The logarithmic regression model that fits these data is $\text{\hspace{0.17em}}y=141.91242949+10.45366573\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$
2. If sales continue at this rate, about 171,000 games will be sold in the year 2015.

## Building a logistic model from data

Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or limiting value . Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients.

It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic regression analysis , we use the form most commonly used on graphing utilities:

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
hi vedant can u help me with some assignments
Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar