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  Theory of Exponential and Logarithmic Models

Exponential and Logarithmic Models:

Exponential Growth:

Function:

2173_1.jpg

y = C ekt, k > 0

Features:

a) Asymptotic to y = 0 to the left.
b) Passes via (0,C).
c) C is the initial value.
d) Increases devoid of bound to right.

Note:

A few things that exponential growth is used to model comprise the population growth, bacterial growth and the compound interest.

When you are lucky adequate to be given the initial value, that is the value if x = 0, then you already know the value of constant C. The only thing essential to complete the model is to have one extra point on the graph. Plug the values for x, y and C, and solve it for k.

Alternatively, almost similar to cheating, you can put the x-values to List 1, the y-values to List 2, and select the ExpReg option on TI-82 calculator.

Exponential Decay (decreasing form):

Function:

452_2.jpg

y = C e-kt, k > 0

Features:

a) Asymptotic to y = 0 to the right.
b) Passes via (0,C).
c) C is the initial value.
d) Reducing, but bounded beneath by y = 0.

Note:

Exponential decay can be used to model the radioactive decay and depreciation.

Exponential decay models reduce very quickly, and then the level off to become asymptotic towards x-axis.

Similar to the exponential growth model, if you know the initial value then rest of the model is fairly simple to complete.

Exponential Decay (increasing form):

Function:

572_3.jpg

y = C (1 - e-kt), k > 0

Features:

a) Asymptotic to y = C to the right.
b) Passes via (0,0).
c) C is the upper limit.
d) Rising, however bounded above by y = C.

Note:

The Exponential decay models of this form can model the sales or learning curves where there is an upper limit. This is completed by subtracting the exponential expression from one and multiplying by upper limit.

Exponential decay models of this form will rise very quickly at first, and then level off to become asymptotic to upper limit.

Similar to the other exponential models, when you know upper limit, then the rest of model is fairly simple to complete.

Calculator will not fit the increasing model including exponential decay directly.

Gaussian Model:

Function:

2170_4.jpg

y = a exp (-(x-c)/b)2

exp(x) is the other way of writing ex

Features:

a) Asymptotic to the x axis to left and right.
b) Passes via (0,a)
c) a controls the height of curve
d) Centered around x = c
e) b controls the spread.
f) Bell shaped.

Note:

The Gaussian model is utilized quite a bit in Statistics to model distribution of scores.

The Gaussian model is introduced by German mathematician Carl Friedrich Gauss and named after them. This is the same Gauss who introduced the Fundamental Theorem of Algebra. This is symmetric about its mean, x = c. To the right of origin, it reduces slowly at first, then more quickly, and then levels off and become asymptotic to x-axis.

Similar to the Exponential Decay model, the Gaussian model can be turned to an increasing function by subtracting the exponential expression from one and then multiplying the upper limit.

Logistics Growth Model:

Function:

2423_5.jpg

y = a/(1 + b e-kx), k > 0

Features:

a) Asymptotic to y = a to the right,
b) Asymptotic to y = 0 to the left,
c) Passes via (0, a/(1+b))
d) Slow growth, followed by the moderate growth and followed by the slow growth.

Note:

The logistics model starts with a slow growth, followed by the period of moderate growth, and then back to the period of slow growth. It consists of an upper limit which can’t be exceeded.

The Logistics model can be employed to approximate the sales and advertising (a little advertising produces a little growth in sales, more advertising produces moderate growth in sales and finally there reaches a point of saturation in which additional advertising advantages little in terms of sales) or the population growth where there is no capacity for unlimited growth (employ the exponential growth model if you require that).

Logarithmic Model:

Function:

1877_6.jpg

y = a + b ln x

Features:

a) Rises without bound to the right.
b) Passes via (1, a).
c) Very fast growth, followed by the slower growth.
d) Common log will grow slower than the natural log.
e) b controls the rate of growth.

The logarithmic model has a period of fast increase, followed by the period where the growth slows, however the growth continues to raise without bound. This makes the model unsuitable where there requires being an upper bound. The major difference between this model and the exponential growth model is that, the exponential growth model starts slowly and then rises very fast as time increases.

Some of the physical applications encompass logarithmic models: The magnitude of earthquakes, intensity of sound and the acidity of a solution.

Earthquakes:

R = log I

The Richter scale is utilized to measure the intensity of earthquake. The real model is a little more complicated, however it simplifies to the equation shown. R is the magnitude on Richter scale of earthquake. I is the intensity of earthquake measured relative to the reference value. That reference value is the minimum seismic activity which can be measured, and consists of the value I0 = 1.

Each and every very rise of 1 in Richter scale signifies the magnitude of earthquake is 10 times greater.

Sound Intensity:

β = 10 log (I/I0)
Sound level, measured in decibels, is given by the formula shown above. The reference value, I0 is the smallest sound intensity that can be heard by the human ear and is roughly equivalent to 1x10-16 watts per square centimeter.

There are 10 decibels to bel. While bel is the real unit, similar to meter, liter or gram, we can use decibel for all practical aims. The rise of 10 decibels is equal to the sound which is 10 times as strong in intensity. The increase of 20 decibels is equal to the sound intensity which is 100 times greater.

Acidity:

pH = - log [ H+]
pH is the measure of acidity of a substance. It varies from 0 to 14, with acids ranging from 0 to 7, 7 being neutral, and the bases ranging from 7 to 14. The [H+] is the concentration of hydrogen ions and is measured in moles per liter. The more is the hydrogen ions, then smaller the pH (note that the negative sign is in front of log) and more acidity the solution is.

Upper Bounds:

Note that the Exponential Growth and Logarithmic models rise without bound to right. The Gaussian and Exponential Decay models both approach to the x-axis to the right. Merely the Logistics Growth model provides you an upper bound to the right. The Exponential Decay and Gaussian models can be made to contain an upper bound by subtracting the exponential expression (therefore making it decreasing rather than increasing) from one and multiplying by the upper bound.

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