P.O.D.-Solve Each Logarithmic Equation ALGEBRAICALLY to Three Decimal Places
P.o.D.-Solve each logarithmic equation ALGEBRAICALLY to three decimal places.
1.) ln(4x)=1
2.) 2-6 ln(x)=10
3.) lnx+2=1
4.) log3x+log3x-8=2
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: y=aebx
2. Exponential decay model: y=ae-bx
3. Gaussian model(bell shaped): y=ae-(x-b)2c
4. Logistic growth model(stretched S): y=a1+be-rx
5. Logarithmic models(upside down hockey stick)
y=a+blnx, y=a+blogx
*Memorize the shapes of these graphs (Fig. 3.29) on page 257.
EX: The population P of a city is given by P=95,300e0.055t, where t=0 represents 1996. According to this model, when did the population reach 150,000?
150,000=95,300e.055t→
1500953=e.055t→
ln1500953=.055t→
8.247=t
Since 1996 is t=0, 8 years later would be 2004.
Exponential Growth: y=aebx
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).
125=ae2b and 350=ae4b
Now solve one of the two equations for a.
125e2b=a
Next, substitute this into the other equation and solve for b.
350=125e2be4b→
350=125e4be2b→
350=125e4b-2b→
350=125e2b→
145=e2b
→ln145=2b→
ln1452=b→
ln14-ln52=b→
.5148≈b
Use this to find a.
a=125e2b=
125e2.5148≈
44.6429
Finally, substitute everything back into the equation for exponential growth.
y=aebx=44.6429e.5148(6)≈979.9439
About 980 fruit flies.
Carbon Dating Model:
R=11012e-t8223
EX: Estimate the age of a newly discovered fossil in which the ratio of carbon 14 to carbon 12 is R=11014.
Substitute.
11014=11012e-t8223→
10121∙11014=e-t8223→
1100=e-t8223→
ln1100=-t8223→-8223ln1100=t→37,868.3144=t
Gaussian Models (Carl Gauss):
1. y=ae-(x-b)2c
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 y=12πe-x22.
EX: Last year, the math scores for students in a particular math class roughly followed the normal distribution given by y=0.0399e-(x-74)2114, 30x110 where x is the math score. Sketch the graph of this function, and use it to estimate the average math score.
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 y=75001+7499e-0.9t, t≥0 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?
y=75001+7499e-0.9(4)=36.4263 or 36 students
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 R=logII0, where I0=1 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. R=log684000001=7.835
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