PRECALCULUS: UNIT 5 NOTES€¦ · Definition of the Logarithmic Function: For > r and 𝑏> r,...

PRECALCULUS: UNIT 5 NOTES

NAME

BLOCK

Learning Target Number

Chapter and Section Title Description

LT1 3.1 Exponential Functions and their Graphs

Sketch graphs of exponential functions with transformations.

LT2 3.1 Exponential Functions and their Graphs

Apply compound interest formulas to solve real-world problems.

LT3 3.2 Logarithmic Functions and their Graphs

Evaluate logarithmic expressions of different bases without a calculator.


Rewrite exponential equations as logarithmic equations, and vice versa.


Evaluate logarithmic expressions of different bases with a calculator.


Identify characteristics of exponential and logarithmic functions from their graphs.


Sketch graphs of logarithmic functions with transformations.

LT8 3.3 Properties of Logarithms Apply logarithmic properties to expand and condense logarithmic expressions.

LT9 3.4 Exponential and Logarithmic Equations

Solve exponential and logarithmic equations algebraically.

LT10 3.5 Exponential Growth and Decay Apply exponential growth and decay formulas to solve contextual problems

Learning Targets 1-2 Exponential Functions Chapter 3.1 and their Graphs

Unit 5 Notes Page 1

Introduction to exponential functions 1. Definition of Exponential Function: 2. Which of these functions are exponential? Why or why not?

y = 2x y = x2 𝑦 = (1

2)

𝑥−1

y = 1x y = 3x+1 y = xx

3. Characteristics of Exponential Functions of the Form f (x) = bx

Domain: Range: Common point: Growth: Decay: Asymptote: Function? One to One?: Sketch graphs of exponential functions with transformations – LT1 4. Sketch the following exponential functions. Plot at least 3 points and identify the asymptote.

a. f (x) = 2x b. f (x) = -2x

Asymptote: __________ Asymptote: __________

Transformations: Transformations:


Unit 5 Notes Page 2

c. f (x) = 2-x d. 𝑓(𝑥) = (1

2)

𝑥



e. 𝑓(𝑥) = (1

2)

𝑥−1 f. f (x) = 2x -1




Unit 5 Notes Page 3

Apply compound interest formulas to solve real-world problems – LT2 Compound Interest Formulas: For n compoundings per year: For continuous compounding: 6. Given: P = $1000, r = 10%, t = 10 years. Find the interest if it is: a. compounded quarterly b. compounded continuously

7. The population of a town increased according to the model P(t) = 100e0.2197t

where t is the time in years ( t = 0 corresponding to 1990). Use the model to predict the population in the following years: a. 1995 b. 2000 c. When will the population be 20,000?

Learning Targets 3-7 Logarithmic Functions Chapter 3.2 and their Graphs

Unit 5 Notes Page 4

Introduction to logarithmic functions The logarithmic function is the inverse of the exponential function. Definition of the Logarithmic Function:

For 𝑥 > 0 and 𝑏 > 0, 𝑏 ≠ 1, y = logb x is equivalent to by = x .

The function f x( ) = logb x is the logarithmic function with base b.

**Common Logs have a base of 10. log10 x = log x( )

**Natural Logs have a base of e. loge x = ln x( )

Evaluate logarithmic expressions of different bases without a calculator – LT3 Rewrite in exponential form and determine the value of x.

1. log216 = x 2. log2

1

8= x 3. log3 27 = x 4. log27 9 = x

5. loga1

a= x 6. ln1 = x 7. logx 64 = 3 8. logx 81 = 4

Rewrite exponential equations as logarithmic equations, and vice versa – LT4 Rewrite in logarithmic form.

9. 82 = 64 10. 9

3

2 = 27

11. 10-3 = 0.001 12. 2-4 =1

16


Unit 5 Notes Page 5

Properties of logarithmic functions Logarithmic Properties Involving 1: logb b logb1

log10 10 log1

lne ln1

Evaluate: 13. log8 8 = _______ 14. log12 1= _______ Inverse Properties of Logarithms:

logb bx blogb x

log10x 10log x

lnex eln x

Evaluate:

15. log4 45= _______ 16. 6log6 9= _______

Evaluate logarithmic expressions of different bases with a calculator – LT5 The logarithmic function will help us understand diverse phenomena, including earthquake intensity, human memory, and the pace of life in large cities. Modeling the Height of Children

The percentage of adult height attained by a boy who is x years old can be modeled by

f (x) = 29 + 48.8 log x +1( )where x represents the boy’s age and f (x)represents the percentage

of his adult height.

17. Approximately what percentage of his adult height has a boy attained at age 8? 18. Approximately what percentage of his adult height has a boy attained at age 10?


Unit 5 Notes Page 6

Identify characteristics of exponential and logarithmic functions from their graphs – LT6

Sketch the graph of y = 2xand its inverse on the same set of axes. What is the inverse of y = 2x ?

Characteristics of Logarithmic Functions Characteristics of Exponential Functions

of the form f (x) = logb x . of the form f (x) = bx .

Domain: Domain: Range: Range: x – intercept: x – intercept: y – intercept: y – intercept: If b > 1, then: If b > 1, then: If 0 < b < 1, then: If 0 < b < 1, then: Asymptote: Asymptote: **Keep in mind the graphs should be smooth and continuous (no sharp corners or gaps).


Unit 5 Notes Page 7

Sketch graphs of logarithmic functions with transformations – LT7

Example: Sketch the graph of y = log3 x +1

Step 1: Identify the inverse (exponential) parent function. [ y = 3x ]

Step 2: Create a t–chart with ordered pairs

that lie on the inverse (exponential) parent

function y = 3x . Choose x values of

-2,-1,0,1,2 .

Step 3: Reverse these ordered pairs to

identify the points that lie on the original

(logarithmic) parent function, 𝑦 = log3 𝑥

Step 4: Identify any transformations. Step 5: Create new t–charts showing each transformation (stretch & reflect first, then shift!) Step 6: Identify and sketch the asymptote. Step 7: Plot the final points and sketch the function.


Unit 5 Notes Page 8

Sketch the graph of the following logarithmic functions. Identify the parent inverse function, transformations, domain, range and asymptote. 18. y = 2log3 x +1

Inverse parent__________ Transformations: Domain_________ Range__________ Asymptote_______

19. y = - log4 x - 2( )

Inverse parent__________

Transformations:

Domain_________

Range__________

Asymptote_______


Unit 5 Notes Page 9

20. y = - log2 -x - 2( )

Inverse parent__________

Transformations:

Domain_________

Range__________

Asymptote_______

Learning Target 8 Properties of Logarithmic Functions Chapter 3.3

Unit 5 Notes Page 10

Properties of Logarithmic Expressions Let 𝑀, 𝑁, 𝑝, and 𝑏 be real numbers such that 𝑏 > 1 and 𝑀, 𝑁 > 0.

Product Rule: log𝑏(𝑀𝑁) = log𝑏(𝑀) + log𝑏(𝑁) Quotient Rule: log𝑏 (𝑀

𝑁) = log𝑏(𝑀) − log𝑏(𝑁)

Power Rule: log𝑏(𝑀𝑝) = 𝑝 ∙ log𝑏(𝑀) Change of Base: log𝑏 𝑀 =log(𝑀)

log (𝑁)

Warning: log𝑏(𝑀 + 𝑁) ≠ log𝑏(𝑀) + log𝑏(𝑁). Be careful!

Apply logarithmic properties to expand and condense logarithmic expressions – LT8

Use the properties of logarithms to expand each expression as much as possible.

1. log7 (19

𝑥) 2. log5(74)

3. ln(√𝑥) 4. log(4𝑥5)

5. ln (𝑒3

7) 6. log𝑏(𝑥3√𝑦)

7. log6 (√3𝑥33

36𝑦4)

Learning Target 8 Properties of Logarithmic Functions Chapter 3.3


Use the properties of logarithms to condense each expression into a single logarithm with a

coefficient of 1.

8. 1

2log(𝑥) + 4 log(𝑥 − 1) 9. 3 ln(𝑥 + 7) − ln (𝑥)

10. 4 log𝑏(𝑥) − 2 log𝑏(6) −1

2log𝑏(𝑦)

11. 3[log𝑎(2) + log𝑎(𝑥)] + log𝑎(4) + log𝑎(𝑥) − log𝑎(𝑥)

Let log𝑎(2) = 𝐴 and log𝑎(5) = 𝐵. Write each expression in terms of A and B.

12. log𝑏 (2

5) = 13. log𝑏(50)

14. log𝑏(16) = 15. log𝑏 (√25

4)

Learning Target 9 Exponential and Logarithmic Equations Chapter 3.4


Solving an Exponential Equation Method 1 (Using Logarithms to Solve):

Isolate the exponential expression; take the logarithm on both sides of the equation; simplify

using the power rule; solve for the variable.

Method 2 (Expressing Each Side as a Power of the Same Base):

Rewrite equation in the form 𝑏𝑀 = 𝑏𝑁 ; set 𝑀 = 𝑁; solve for the variable.

Solving a Logarithmic Equation

Method 1 (Using the Definition of Logarithm):

Rewrite the equation in exponential form; solve for the variable.

Method 2 (Using the One – to – One Property of Logarithms):

Rewrite the equation in the form log𝑏(𝑀) = log𝑏(𝑁); set 𝑀 = 𝑁; solve for the variable.

Solve exponential and logarithmic equations algebraically – LT9

1. − 14 + 3𝑒𝑥 = 11 2. 23−𝑥 = 565 3. 3000

2 + 𝑒2𝑥= 2

4. 4𝑒2𝑥 − 8𝑒𝑥 − 5 = 0 5. 1

4(3)−𝑥 − 18 = 18 6. 15 + 2 log(𝑥) = 31

Learning Target 9 Exponential and Logarithmic Equations Chapter 3.4


7. 4 ln(3𝑥) = 18 8. ln(𝑥 + 2) + ln(𝑥) = 2

9. log4(𝑥) − log4(𝑥 − 1) =1

2 11. log3(80𝑥2) − log3(𝑥2 − 1) = 4

11. Medical research indicates that the risk of having a car accident increases exponentially as

the concentration of alcohol in the blood increases. The risk is modeled by𝑅 = 6𝑒12.77𝑥, where x

is the blood alcohol concentration and R, given as a percent, is the risk of having a car accident.

What blood alcohol concentration corresponds to a 20% risk of a car accident?

12. How long will it take $1000 to grow to $3600 at 8% interest compounded quarterly?

Remember: 𝐴 = 𝑃 (1 +𝑟

𝑛)

𝑛𝑡

Learning Target 10 Exponential Growth and Decay Chapter 3.5


Apply exponential growth and decay formulas to solve contextual problems – LT10

Use the compound growth formula, 𝐴 = 𝑃𝑒𝑟𝑡 , to solve the following. 1. An investment of $8,000 is compounded continuously. What annual percentage rate will

produce a balance of $30,000 in 10 years?

2. The half-life of radioactive iodine is 60 days (After 60 days, a given amount of radioactive

iodine will have been decayed to half of its original amount). Suppose a contained nuclear

accident occurs and gives off an initial amount A0 of radioactive iodine.

a. Write an equation for the amount of radioactive iodine present at any time t following the

accident. (Hint: find the decay rate, k)

b. How long will it take for the radioactive iodine to decay to a level of 20% of the original

amount? (Hint: use the decay rate found in part a)

3. Radium 226 has a half-life of 1620 years.

a. Find the general equation for the decay of the isotope. (Hint: find the decay rate, k)

b. Find the amount left after 1000 years if you start with 5 grams.

Learning Target 10 Exponential Growth and Decay Chapter 3.5


4. The population of Houston, Texas was 1.63 million in 1990 and its population for the year

2010 was 2.1 million. Write the exponential growth equation for the population growth of

Houston by letting t = 0 correspond to 1990. Use the model to predict the population of the city

in 2020.

Logistic Growth Model – Modeling the Spread of the Flu 5. On a college campus of 5000 students, one student returned from vacation with a contagious flu virus.

The spread of the virus through the student body is given by:

𝑓(𝑡) =5000

1 + 4999𝑒−.08𝑡, where 𝑓(𝑡) is the total number of students infected after 𝑡 days.

The college will cancel classes when 40% or more of the students are ill.

a. Use a graphing utility to graph the function and determine the asymptote.

b. How many students are infected after 5 days? c. After how many days will the college cancel classes?

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PRECALCULUS: UNIT 5 NOTES€¦ · Definition of the Logarithmic Function: For > r and 𝑏> r,...

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