Central Valley Christian Schools

Honors Precalculus Chapter 11 Page 1

Section 11.1 – Rational Exponents Goals: 1. To use the properties of exponents. 2. To evaluate and simplify expressions containing rational exponents. I. Properties to Review

A. m na a = B. ( )nma =

C. na

b⎛ ⎞ =⎜ ⎟⎝ ⎠

D. ( )mab =

E. m

n

aa

=

F. ma− =

G. 1ma−=

H. 0a =

I. 1na =

J. mna =

II. Examples A. Simplify:

1. ( ) ( )3 22 3x x−− ⋅

2. 7 5 7 55 5 5 5− −⋅ ⋅ ⋅ 3. 4 216 4. 5 3144x y

5. 238

B. Express using Radicals

1. 32y

2. 1 3 52 2 22 a b

3. 2 33 2x y

4. ( )11

2 53s t v

Homework: p. 700 – 1, (6 – 63)/3, 65-68 all, 71,73, 75-83 all


Section 11.2 – Exponential Functions Goals: 1. To evaluate expressions with irrational exponents. 2. To graph exponential functions. 3. To graph exponential inequalities. I. Graphing Exponential Functions

A. The function: xy a b c= ⋅ + 1. a represents

a) b)

2. b represents a) b)

3. c represents –

B. Graph Manually 1. 13 2xy −= ⋅ 2. 3 2 4xy ≤ − ⋅ +

II. Examples 1. Suppose your parents deposited a one time amount of $5400 in an account paying 12% APR. Suppose the account paid interest monthly. Find the account balance after 18 years. 2. The average growth rate of the population of a city is 7.5% per year and the city’s population is 22,750. What is the expected population in 10 years?

Homework: p. 708 – 1-4 all, 10-14 all, 18-21 all, 23-29 all, 31-43 all

3 2xy = ⋅


Section 11.3 – The Number e Goals: 1. To use the exponential function xy e= I. The Natural Number, e

A. 2.7182818....e = B. When it comes to the physical world e is only natural! J

II. Some Natural Uses A. Statistics: Graph

2xy e−= [-2,2]x[-1,2] B. Finance: An investment compounded continually rtA Pe= C. Science: Newton’s Law of Cooling kty ae c−= +

D. Construction: catenary curves 2

x xe ey−+

=

E. Biology: Ebbinghaus Model ( )100 btp a e a−= − + F. Population: rtP Ce=

Homework: p. 714 – 1, 2, 4, 8-15 all, 17, 18, 20-25 all, 27, 28


Section 11.4 – Logarithmic Functions Goals: 1. To evaluate expressions involving logarithms. 2. To solve equations involving logarithms. 3. To graph logarithmic functions and inequalities. I. Logarithms

A. Inverse function of exponential functions xy a= B. Notation: If xy a= , then loga y x= . C. Similarities:

1. If siny x= , then 1sin y x− = 2. If 2y x= , then y x= 3. If 3y x= + , then 3y x− =

D. Question asked for loga b : “a to what power will equal b?” E. Equivalence Statements: logx

ay a y x= ⇔ = F. Examples:

1. Write the following in exponential form

a) 81log 23

= b) 10log 2x =

2. Write the folowing in logarithmic form a) 5 yx= b) ( )18 yx −=

3. solve the following a) 7log ( 3) 2x− = b) 2log 8 x=

II. Graphing Logarithms Examples

1. 3logy x= 2. 2logy x=

2logy x=

2xy =


III. Logarithmic Properties A. Properties:

1. logb mn = 3. log pb m =

2. logbmn= 4. loglog or b mm

b b b= =

B. Examples: Expand the following logarithms.

1. 2log xy 3. 3

2log xy

2. 2log xy

4. 2 32log x y

C. Examples: Write the following as a single logarithm 1. 2 2 2log log logx y z+ − 3. ( )2 2 23 log log 2log 5y z+ − 2. 2 23log 2logz x+ 4. 2 2 22log 5log 3log 3y x− +

D. Examples: Given 2 2log 3 and log 2x y= =

1. Find: 2log xy 4. Find: 3

2log xy

2. Find: 2log xy

5. Find: 2 32log x y

3. Find: 2log 32x 6. Find: 2

2 5

16log xy


E. Solve 1. 8 84log log 81x = 2. 3 3log (2 5) log (5 4)x x+ = − 3. 3 3log (4 5) log (3 2 ) 2x x+ − − = 4. 2

5 5 5log ( 3) log ( 1) log 7x x+ − − = 5. If a bacterial colony doubles every 20 minutes and there are 500 bacteria at noon, when will the population reach 16,000? (do this two ways)

Homework: p. 722 – 1, 4, 6-16 all, 21-51 odds, 53-57 odds, 59, 62, 65-78 all (exclude 70) and Worksheet


Honors Precalculus Worksheet Section 11.4

NO CALCULATOR ON THIS WORKSHEET Name: ______________________________ For each of the following: Part A: For problems 1 – 8, expand each logarithm completely. Part B: Given log 2x a = , log 7x b = , log 3x c = − , and log 9x d = − , find value for each logarithm.

1. ( )logx ab A: B:

2. logxbc⎛ ⎞⎜ ⎟⎝ ⎠

A: B:

3. 2log x c A: B:

4. logbabc

⎛ ⎞⎜ ⎟⎝ ⎠

A: B:

5. ( )2logx a bc A: B:

6. 2

2logxdc

⎛ ⎞⎜ ⎟⎝ ⎠

A: B:

7. 3 2

logxa bd

⎛ ⎞⎜ ⎟⎝ ⎠

A: B:

8. 2

3log xadc b

⎛ ⎞⎜ ⎟⎝ ⎠

A: B:

9. 3log x x B:

10. log 7xx B:

11. ( )3 2logx x a B:

12. 5

logxx ba

⎛ ⎞⎜ ⎟⎝ ⎠

B:


Write the following as a single logarithm. 13. 4 42log logx z− 14. 3 3 3log 3log 2loga b c− +

15. ( )3 3 3 33 log log 5log 2log 5a c b− + − 16. ( ) ( )7 7log 2 log 3x x+ + −

17. ( ) ( )2 2log 5 2log 2x x+ − − 18. ( ) ( ) ( )5 5 5log 8 5 log 2 log 2x x x+ − − + +


Section 11.5 – Common Logarithms Goals: 1. To find common logarithms and antilogarithms. 2. To solve problems involving common logarithms.

A. Common logarithms 1. The common logarithm is 10log 2. log 1000 = 3. log 100 = 4. log 0.01 = 5. log 27 =

B. Converting logarithms: logb a =

1. Find 3log 7 2. Find log 2e

C. Antilogarithms

1. Solve: 3 log x= 2. Solve: 0.2568 log x=

D. Applications 1. The annual growth rate of the moose population in Wells Gary Park in northern British

Columbia, Canada, is 6.7%. This situation can be modeled by the equation log

log1.067

T

i

PPn = ,

where PT = population today, Pi = initial population, and n = time in years. A few years ago the population was 325 moose and today it is 450 moose. Find the number of years that have elapsed between the initial population and today’s population.

2. The pH of a solution is related the concentration of hydrogen ions it contains by the

formula 1pH logH +

= , where H + is the number of gram ions of hydrogen ions per liter. If

the pH of ocean water is 8.8, what is the concentration of hydrogen ions?

Homework: p. 730 – 1, 28-39 all, 52-54 all, 58a, 60a, 62a, 65-76 all (exclude 75)


Section 11.6 – Natural Logarithms Goals: 1. To find natural logarithms. 2. To solve problems involving natural logarithms.

A. Natural logarithms 1. The natural logarithm is loge 2. ln e = 3. 6ln e = 4. ln1= 5. ln 2 = 6. ln 3.457 =

B. Converting logarithms: logb a =

1. Find 2log 7 2. Find 7log 2

C. Solve

1. *3 ln x= 2. *0.2568 ln x= 3. 510 5 xe= 4. 0.32ln 23 ln xe=

Homework: p. 735 – 1, 3, 18-35 all, 62-70 all (exclude 68)


Section 11.6/5B – Solving Exponential Function Goals: 1. To solve exponential and exponential equations. 2. To solve exponential and exponential inequalities. Examples 1. Solve two ways: 13.9 28x+ =

• With common logs:

• With natural logs:

2. Phosphorus-32 is a radioactive substance with a half-life of 14.3 days. How long will it take to reduce a 100-gram sample of P-32 to 15 grams?

3. Under ideal conditions, the population of a certain bacterial colony will double in 45 minutes. How much time will it take for the population to increase five-fold?

4. Solve: 24 9x x−=


5. The exponential function y x= ⋅215 0955. ( . ) models the amount of whole milk each person in the United States consumers in a year. (y is the number of gallons of whole milk and x is the number of years since 1975.) Which year, month, and day did whole milk consumption fall to 10.8 gal/person?

6. Solve by graphing (a) 1 22 5x x− −= (b)

2 14 12x − < Homework: p. 730 – 41-51 odds, 55, 57, 58b, 60b, 62b p. 736 – 37-53 odds, 54 -60 all

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