P.o.D.-Solve each logarithmic equation ALGEBRAICALLY to three decimal places. 1.) ln(4x)=1 2.) 2-6 ln(x)=10 3.) 4.)

1.) 0.6796

2.) 0.2636

3.) 5.39

4.) 9 3.5 – Exponential and Logarithmic Models *Memorize the five most common types of mathematical models involving exponential and logarithmic functions (pg.257) 1. Exponential growth model: 2. Exponential decay model: 3. Gaussian model(bell shaped): 4. Logistic growth model(stretched S): 5. Logarithmic models(upside down hockey stick) *Memorize the shapes of these graphs (Fig. 3.29) on page 257. EX: The population P of a city is given by , where t=0 represents 1996. According to this model, when did the population reach 150,000?

Since 1996 is t=0, 8 years later would be 2004. Exponential Growth: EX: In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. After 2 days there are 125 flies, and after 4 days there are 350 flies. How many flies will there be after 6 days? Substitute what we know into the equation(s).

Now solve one of the two equations for a.

Next, substitute this into the other equation and solve for b. Use this to find a. Finally, substitute everything back into the equation for exponential growth.

About 980 fruit flies. Carbon Dating Model:

EX: Estimate the age of a newly discovered fossil in which the ratio of carbon 14 to carbon 12 is . Substitute. Gaussian Models (Carl Gauss): 1.

2. Used with “normally” distributed data

3. Will be a bell shaped curve

4. The average value of a population will occur at the maximum of the function. Side Note: A standard normal distribution fits the model . EX: Last year, the math scores for students in a particular math class roughly followed the normal distribution given by , 30

The average score for students in the math class was 74. EX: On a college campus of 7500 students, one student returns from vacation with a contagious and long- lasting virus. The spread of the virus is modeled by where y is the total number of students affected after t days. The college will cancel classes when 30% or more of the students are affected. How many students will be infected after 4 days?

After how many days will the college cancel classes? 30% of 7500 is 2250. Graph the equation and y=2250

about 9 days EX: On the Richter scale, the magnitude R of an earthquake of intensity I is given by , where is the minimum intensity used for comparison. Find the magnitude R of an earthquake of intensity I.

a. I=68,400,000

b. I=42,275,000 a.

b. R=log(42275000)=7.626 Upon completion of this lesson, you should be able to:

1. Identify the different types of exponential models by their equation and graph.

2. Solve story problems involving the different exponential models. For more information, visit http://academics.utep.edu/Portals/1788/CALCULUS %20MATERIAL/3_5%20EXPO%20AND%20LOG %20MODELS.pdf

HW Pg.264 3-30 3rds, 40, 43, 63, 73- 78 Quiz 3.3-3.5 tomorrow