algebra 2 and trigonometry honors

1

Algebra 2 and

Trigonometry

Honors

Chapter 8: Logarithms – Part A

Name:______________________________

Teacher:____________________________

Pd: _______

2

Table of Contents

Day 1: Inverses and Graphs of Logarithmic Functions & Converting an Exponential Equation into a

Logarithmic Equation SWBAT: Convert an Exponential Equation into a Logarithmic Equation & Graph Logarithmic Functions

Pgs. 3 – 9 in Packet

HW: 10 – 11 in Packet

Day 2: “e” and The Natural Log

SWBAT: Learn the properties of “e” and the natural log Pgs. 12– 17 in Packet


Day 3: Properties of Logs

SWBAT: Learn the properties of logs. Pgs. 22– 26 in Packet


Day 4: “REVIEW DAY” SWBAT: Review all the log properties

Pgs. 29– 32 in Packet


QUIZ on Day 5

***Answer Keys start at page 33****

1

Day 1 – Introduction into Logarithmic Functions

Warm - Up:

Using the table below: a) Complete the table of values for y= 2x

b) sketch the graph of y= 2x

x y

-2

-1

0

1

2

2) Recall:

How do we find the inverse of a function? ___________________________________________

Find the inverse algebraically.

3) Graph the inverse of the function y = 2x.

Properties of

Properties of

Domain:

Domain:

Range:

Range:

Asymptote:

Asymptote:

x-intercept:

x-intercept:

y-intercept:

y-intercept:

2

Example 1: Graph 𝑓(𝑥) = (1

3)

𝑥

and its inverse 𝑓−1(𝑥).

Ex 2: Graph 𝑦 = log5 𝑥 and its inverse and its inverse 𝑓−1(𝑥).

Concept 1:

3

Regents Questions 3)

4)

5) The graph of the function 𝑦 = 4𝑥 appears in which two quadrants?

(1) I and II (2) I and IV (3) II and III (4) III and IV

6) The graph of the function 𝑦 = log4 𝑥 appears in which two quadrants?

(1) I and II (2) I and IV (3) II and III (4) III and IV

7) The inverse of the function 𝑦 = log25 𝑥

(1) 𝑦25 = 𝑥 (2) y = 25𝑥 (3) x = 25𝑦 (4) 𝑦 = log𝑥 25

8) The inverse of the function 𝑓(𝑥) = 15𝑥

(1) 𝑓−1(𝑥) = 𝑥15 (2) 𝑓−1(𝑥) = log𝑥 15 (3) 𝑓−1(𝑥) = log15 𝑥 (4) 𝑓−1(𝑥) = log15 𝑦

4

Concept 2: Converting from Exponential to Logarithmic Form and Vice Versa

Until now, there was no way to isolate y in an equation of the form 𝑥 = 3𝑦. You cannot take the “yth root” of something if that something isn’t a value. The word “Logarithm” means “power.” When you see the function “log,” you should translate that into “the power I raise…” Example: Log2 8 = 3

Example: X = Log10 1000

So,

=

The power I raise 2 to to get

8 is 3 What 10000

The power I raise 10 to to get

is

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Concept 2: Simplifying a Log Expression “What power?????"

Teacher Modeled Student Try it!

𝐥𝐨𝐠𝟓 𝟐𝟓

𝐥𝐨𝐠𝟏𝟎 𝟏𝟎, 𝟎𝟎𝟎

𝐥𝐨𝐠𝟒 𝟏

𝐥𝐨𝐠𝟓 𝟔𝟐𝟓

𝐥𝐨𝐠𝟔 (𝟏

𝟑𝟔)

𝐥𝐨𝐠𝟓 √𝟓

Regents Question Simplify

a) Simplify: 7log7 4

b) Simplify: log5 125𝑥

6

Concept 3: How do we change between log form and exponential form?

Teacher Modeled Student Try it!

𝐥𝐨𝐠𝟖 𝒙 = 𝟏

𝟑

𝐥𝐨𝐠𝟑𝟐 𝒙 = 𝟐

𝟓

𝐥𝐨𝐠𝒙 𝟑𝟔 = 𝟐

𝐥𝐨𝐠𝒙(−𝟐𝟏𝟔) = 𝟑

𝐥𝐨𝐠𝒙 𝟏𝟐𝟓 = −𝟑

𝟐

𝐥𝐨𝐠𝒙 𝟐𝟒𝟑 = −𝟓

𝟐

7

Concept 4: How Do we evaluate Common Logs?

Base with NO Exponent Radical with NO index

X √𝟐𝟓 means……. means…….

Logarithms with a base Logarithms with NO base

𝐥𝐨𝐠𝟓 𝒙 = 𝟐 𝐥𝐨𝐠 𝟏𝟎𝟎𝟎 means……. means…….

On the graphing Calculator

8

Evaluate each.

𝐥𝐨𝐠 𝟏𝟎𝟎

Explanation

𝐥𝐨𝐠 (𝟏

𝟏𝟎, 𝟎𝟎𝟎)

𝐥𝐨𝐠 𝟏𝟎𝟎

𝐥𝐨𝐠 𝟏𝟎

𝐥𝐨𝐠 𝟏𝟕

𝟐𝐥𝐨𝐠 𝟒

9

Challenge: Solve the following Equation

Summary/Closure:

Exit Ticket:

10

Day 1 – HW

11

27. Sketch below the graph of .

a) State the domain and range of the graph.

b) Write the equation of the asymptote.

28.

12

Day 2: “e” and the Natural Log Warm – Up

1.

2. Fdf

3.

4.

13

There are many numbers in mathematics that are more important than others because they find so many

uses in either mathematics or science. Good examples of important numbers are 0, 1, i, and . In this

lesson you will be introduced to an important number given the letter e for its “inventor” Leonhard Euler

(1707-1783). This number plays a crucial role in Calculus and more generally in modeling exponential

phenomena.

Exercise #1: Which of the graphs below shows xy e ? Explain your choice. Check on your calculator.

(1) (2) (3) (4)

Explanation:

Because of the importance of xy e , its inverse, known as the natural logarithm, is also important.

The natural logarithm, like all logarithms, gives an exponent as its output. In fact, it gives the power that

we must raise e to in order to get the input.

Exercise #2: Without the use of your calculator, determine the values of each of the following.

(a) ln e (b) ln 1 (c) 5ln e (d) ln e

THE NUMBER e

1. Like , e is irrational. 2. e 3. Used in Exponential Modeling

y

x

y

x

y

x

y

x

THE NATURAL LOGARITHM

The inverse of :

14

Exercise #3:

Exercise # 4:

Exercise #5: On the grid below, the solid curve represents xy e . Which of the following exponential

functions could describe the dashed curve? Explain your choice.

(1) 12

x

y (3) 2xy

(2) xy e (4) 4xy

Exercise #6: Determine the domain of the function 2log 3 4y x . State your answer in set-builder

notation.

y

x

15

Exercise #7: Identify the domain and range of each. Then sketch the graph.

16

Exercise #8: A hot liquid is cooling in a room whose temperature is constant. Its temperature can be modeled

using the exponential function shown below. The temperature, T, is in degrees Fahrenheit and is a function of

the number of minutes, m, it has been cooling.

0.03101 67mT m e

(a) What was the initial temperature of the water at

0m . Do without using your calculator.

(b) How do you interpret the statement that

60 83.7T ?

(c) Using the natural logarithm, determine

algebraically when the temperature of the liquid

will reach 100 F . Show the steps in your

solution. Round to the nearest tenth of a minute.

(d) On average, how many degrees are lost per

minute over the interval 10 30m ? Round to

the nearest tenth of a degree.

17

Summary

Exit ticket

18

Day 2 – HW

1. The domain of 3log 5y x in the real numbers is

(1) | 0x x (3) | 5x x

(2) | 5x x (4) | 4x x

2. Which of the following equations describes the graph shown below?

(1) 5logy x (3)

3logy x

(2) 2logy x (4)

4logy x

3. Which of the following represents the y-intercept of the function 2log 32 1y x ?

(1) 8 (3) 1

(2) 4 (4) 4

4. Which of the following values of x is not in the domain of 5log 10 2f x x ?

(1) 3 (3) 5

(2) 0 (4) 4

5. Which of the following is true about the function 4log 16 1y x ?

(1) It has an x-intercept of 4 and a y-intercept of 1 .

(2) It has x-intercept of 12 and a y-intercept of 1.

(3) It has an x-intercept of 16 and a y-intercept of 1.

(4) It has an x-intercept of 16 and a y-intercept of 1 .

y

x

19

6. Determine the domains of each of the following logarithmic functions. State your answers using any

accepted notation. Be sure to show the inequality that you are solving to find the domain and the

work you use to solve the inequality.

(a) 5log 2 1y x (b) log 6y x

7. Which of the following is closest to the y-intercept of the function whose equation is 110 xy e ?

(1) 10 (3) 27

(2) 18 (4) 52

8. On the grid below, the solid curve represents xy e . Which of the following exponential functions

could describe the dashed curve? Explain your choice.

(1) 12

x

y (3) 2xy

(2) xy e (4) 4xy

9. Which of the following values of t solves the equation 25 15te ?

(1) ln15

10 (3) 2ln3

(2) 1

2ln 5 (4)

ln 3

2

10. At which of the following values of x does 22 32xf x e have a zero?

(1) 5

ln2

(3) ln8

(2) ln 4 (4) 2

ln5

y

y

x

20

13. For the equation ctae d , solve for the variable t in terms of a, c, and d. Express your answer in terms

of the natural logarithm.

21

APPLICATIONS

14. Flu is spreading exponentially at a school. The number of new flu patients can be modeled using the

equation 0.1210 dF e , where d represents the number of days since 10 students had the flu.

(a) How many days will it take for the number of new flu patients to equal 50? Determine your

answer algebraically using the natural logarithm. Round your answer to the nearest day.

(b) Find the average rate of change of F over the first three weeks, i.e. 0 21d . Show the

calculation that leads to your answer. Give proper units and round your answer to the nearest

tenth. What is the physical interpretation of your answer?

22

Day 3 – Properties of Logarithms

26

Challenge

Summary/Closure:

Exit Ticket:

27

Day 3 - HW

29

Day 4: Review of Logarithms (Days 1 – 3)

31

Common Core Problems Sets

33

Answer Keys Day 1

27. y = 𝐥𝐨𝐠 𝒙

𝐥𝐨𝐠 𝟒 28.

x y

0 error

1 0

2 .5

4 1

8 1.5

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Day 2

7.

8.

35

7.

8.

9.

10.

11. C 12. A

13.

36

14.

37

Day 3

39

Day 4: Common Core Problem Sets

algebra 2 and trigonometry honors

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