# 6.8 Fitting exponential models to data  (Page 3/12)

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 Month 1 2 3 4 5 6 7 8 Debt ($) 620 761.88 899.8 1039.93 1270.63 1589.04 1851.31 2154.92 1. Use exponential regression to fit a model to these data. 2. If spending continues at this rate, what will the graduate’s credit card debt be one year after graduating? 1. The exponential regression model that fits these data is $\text{\hspace{0.17em}}y=522.88585984{\left(1.19645256\right)}^{x}.$ 2. If spending continues at this rate, the graduate’s credit card debt will be$4,499.38 after one year.

Is it reasonable to assume that an exponential regression model will represent a situation indefinitely?

No. Remember that models are formed by real-world data gathered for regression. It is usually reasonable to make estimates within the interval of original observation (interpolation). However, when a model is used to make predictions, it is important to use reasoning skills to determine whether the model makes sense for inputs far beyond the original observation interval (extrapolation).

## Building a logarithmic model from data

Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the way they increase or decrease that helps us determine whether a logarithmic model is best.

Recall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By reflecting on the characteristics we’ve already learned about this function, we can better analyze real world situations that reflect this type of growth or decay. When performing logarithmic regression analysis , we use the form of the logarithmic function most commonly used on graphing utilities, $\text{\hspace{0.17em}}y=a+b\mathrm{ln}\left(x\right).\text{\hspace{0.17em}}$ For this function

• All input values, $\text{\hspace{0.17em}}x,$ must be greater than zero.
• The point $\text{\hspace{0.17em}}\left(1,a\right)\text{\hspace{0.17em}}$ is on the graph of the model.
• If $\text{\hspace{0.17em}}b>0,$ the model is increasing. Growth increases rapidly at first and then steadily slows over time.
• If $\text{\hspace{0.17em}}b<0,$ the model is decreasing. Decay occurs rapidly at first and then steadily slows over time.

## Logarithmic regression

Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time. We use the command “LnReg” on a graphing utility to fit a logarithmic function to a set of data points. This returns an equation of the form,

$y=a+b\mathrm{ln}\left(x\right)$

Note that

• all input values, $\text{\hspace{0.17em}}x,$ must be non-negative.
• when $\text{\hspace{0.17em}}b>0,$ the model is increasing.
• when $\text{\hspace{0.17em}}b<0,$ the model is decreasing.

Given a set of data, perform logarithmic regression using a graphing utility.

1. Use the STAT then EDIT menu to enter given data.
1. Clear any existing data from the lists.
2. List the input values in the L1 column.
3. List the output values in the L2 column.
2. Graph and observe a scatter plot of the data using the STATPLOT feature.
1. Use ZOOM [9] to adjust axes to fit the data.
2. Verify the data follow a logarithmic pattern.
3. Find the equation that models the data.
1. Select “LnReg” from the STAT then CALC menu.
2. Use the values returned for a and b to record the model, $\text{\hspace{0.17em}}y=a+b\mathrm{ln}\left(x\right).$
4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data.

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function