**3.5 Exponential and Logarithmic Models**

**5 most common**:

1). Exponential growth model: *y* = *a*e^{(bx)}, *b*>0.

2). Exponential decay model: *y* = *a*e^{(-bx)}, *b*>0.

3). Gaussian model: *y* = *a*e^{(-((x-b)2)/c)}

4). Logistic growth model: *y* = *a*/(1+*b*e^{(-rx)})

5). Logarithmic models:

**a**. *y* = *a* + *b* ln *x*; **b**. *y* = *a* + *b* log *x*

Graphs:

Examples:

Exponential Growth: Given P = 240360e^{(0.012t)}

*t* = 0 represents 2000, when will the population reach 275,000?

275000 = 240360 e^{(0.012t)}

1.144117158 = e^{(0.012t)}

ln 1.144117158 = 0.012 *t*

11.21944152 = *t*

Therefore, 2000 + 11 = 2011.

Example 2:

Country: Croatia, in 1997 had 5.0 million people and is estimated to have 4.8 in 2020, how many will there be in 2030?

Two points: (0, 5) and (23, 4.8)

Find the exponential growth model using *y* = *a*e^{(bt)}

5 = *a*e^{(b*0)}

5 = *a*e^{(0)}

5 = *a*

4.8 = *a*e^{(b * 23)}

4.8 = 5e^{(23b)}

.96 = e^(23*b*)

ln .96 = 23*b*

-.0017748693 = *b*

*Croatia’s equation is y = *5e^{(-.0017748693t)}

So in 2030, the population is 4.7 million

**The “b” is the growth rate.**

**Since “b” is negative in Croatia, the population is decreasing.**

**Gaussian Models ***y = a*e^{-((x-b)2)/c}

The model is commonly used in probability and statistics to represent populations that are NORMALLY DISTRIBUTED. For STANDARD normal distributions, the model takes the form:

y = (1/(standard deviation times the square root of 2 pi) )times (e^{((-x^2)/(2 standard deviation squared))})

The graph of a Gaussian model is called a bell-shaped curve.

Example:

The SAT scores in 1997 followed:

y = 0.0036e^{((-(2-511)^2)/(25088))} where 200 is less than or equal to x which is less than or equal to 800.

The maximum point is (511.11112, 0.0036)

**A logistic growth curve:** is a rapid growth followed by a declining rate of growth OR a sigmoidal curve.

y = a / ( 1 + be^{(-rx)})

Example: Spread of a virus (flu)

y = 5000/(1 + 4999e^{(-.8r)})

a. How many students are infected after 5 days?

54 people

b. If classes are cancelled when 40% or more are infected of a campus of 5000 students, how long will it take? 10.14 days.

**Magnitudes of Earthquakes**

**R = log (I/ I _{o})**

**Magnitude R of an earthquake of Intensity of I where I _{0} usually equals 1.**

Example: Find the magnitude of R of an earthquake of Intensity I, let I_{0} = 1.

a). Taiwan in 1999, I = 39811000

R = log 39811000 = 7.600003087 = 7.6

Example 2: What is the Intensity if R = 8.6?

8.6 = log I

10^{(8.6)} = I

I = 398107170.6

**Homework: #30 pg. 266; #1-29 odd, 39**