Grade 12 Pre-Calculus Mathematics

[MPC40S]

Chapter 7

Exponential Functions

Chapter 8

Logarithmic Functions

Outcomes

R7/8, R9, R10

12P.R.7. Demonstrate an understanding of logarithms. 12P.R.8. Demonstrate an understanding of the product, quotient, and power laws of logarithms. 12P.R.9. Graph and analyze exponential and logarithmic functions. 12P.R.10. Solve problems that involve exponential and logarithmic equations.

MPC40S Date: ____________________

Pg. #2

MPC40S Date: ____________________

Pg. #3

Chapter 7/8 – Homework

Section Page Questions

MPC40S Date: ____________________

Pg. #4

MPC40S Date: ____________________

Pg. #5

Chapter 7: EXPONENTIAL FUNCTIONS

7.1 – Characteristics of Exponential Functions Exponential Function: A function of the form _________________________________ where 𝑐 is a constant (𝑐 > 0) and 𝑥 is a variable. Constant → Base Variable → Exponent Let’s look at some graphs of exponential functions. 𝑦 = 𝑐𝑥 where 𝑐 > 1 𝑦 = 𝑐𝑥 where 0 < 𝑐 < 1 Increasing Function Decreasing Function Domain: (−∞, ∞) Domain: (−∞, ∞) Range: (0, ∞) Range: (0, ∞) Asymptote: 𝑦 = 0 Asymptote: 𝑦 = 0 Point on the graph: (0,1) Point on the graph: (0,1)

R9

MPC40S Date: ____________________

Pg. #6

Example #1

Sketch the graph of 𝑓(𝑥) = 2𝑥

Notes:

• The base of this exponential function is ______________

• Thus, the function is _____________________________

• The graph passes through the point _________________

• There is a horizontal asymptote at __________________ (since y cannot be negative)

• The domain is ________________

• The range is _________________

• The graph approaches _______________ when ______________

𝑥 𝑦

MPC40S Date: ____________________

Pg. #7

Example #2

Sketch the graph of 𝑦 = (1

2)

𝑥

Example #3

Sketch the graph of 𝑓(𝑥) = 3𝑥

𝑥 𝑦

MPC40S Date: ____________________

Pg. #8

Example #4

Sketch the graph of 𝑓(𝑥) = 𝑒𝑥

MPC40S Date: ____________________

Pg. #9

Chapter 7: EXPONENTIAL FUNCTIONS

7.2 – Transformations of Exponential Functions

𝑦 = 𝑎 ∙ 𝑐𝑏(𝑥−ℎ) + 𝑘

Example #1

Sketch each of the following exponential functions. a) 𝑓(𝑥) = 2𝑥 + 1 _________________________________ _________________________________ _________________________________

b) 𝑦 = −3𝑥+1 _________________________________ _________________________________ _________________________________ _________________________________

R9

MPC40S Date: ____________________

Pg. #10

c) 𝑔(𝑥) = 4−𝑥 − 1 ______________________________ ______________________________ ______________________________

d) 𝑦 = 31

2𝑥

______________________________ ______________________________ ______________________________

MPC40S Date: ____________________

Pg. #11

Example #2

a) Sketch the graph of 𝑦 = 2(3)𝑥−1 − 4 _______________________________ _______________________________ _______________________________ _______________________________ b) Write the equation of the asymptote. Note: The ‘2’ is a constant and doubles all of the y values.

If 𝑓(𝑥) = (3)𝑥 is the original function, then 𝑦 = 2𝑓(𝑥 − 1) − 4 would be the equation of this function.

MPC40S Date: ____________________

Pg. #12

MPC40S Date: ____________________

Pg. #13

Chapter 7: EXPONENTIAL FUNCTIONS

7.3 – Solving Exponential Equations Before solving exponential equations, let’s review the exponent laws.

(𝑥4)(𝑥3) = __________ (𝑥4)2 = __________ 𝑥15

𝑥3 = __________

___________________ ___________________ ___________________ ___________________ ___________________ ___________________ Big Idea!

If 𝑐𝑥 = 𝑐𝑦 then

For example, 3𝑥+2 = 37 then we can say that

R10

MPC40S Date: ____________________

Pg. #14

Solving Exponential Equations Algebraically When solving exponential equations, we need both sides of the equation to have the same base. Example #1

Solve 2𝑥+1 = 32 Example #2

Solve 34𝑥−1 = 272𝑥

MPC40S Date: ____________________

Pg. #15

Example #3

Solve (3𝑥)(27) = 812𝑥−1 Example #4

Solve 4𝑥+1 = (2𝑥)(√2)

MPC40S Date: ____________________

Pg. #16

Example #5

Solve 1

3𝑥−1 = 81

Example #6

Solve (−8)5

3 = 2 (4𝑥

2)

Note: We cannot yet solve equations with different bases (i.e. 10𝑥+1 = 7𝑥)

MPC40S Date: ____________________

Pg. #17

Chapter 8: LOGARITHMIC FUNCTIONS

8.1 – Understanding Logarithms The logarithmic function is the inverse of the exponential function. Remember, to find the inverse of a function, we switch the 𝑥 and 𝑦 values and then

solve for 𝑦. Exponential Function Inverse Function 𝑦 = 𝑏𝑥 𝑥 = 𝑏𝑦 Notice that the 𝑦 value is now an exponent. Right now, we do not know how to isolate the 𝑦 value! In order to isolate and manipulate exponents, we must use something called the logarithmic function. Logarithmic Function: A function of the form _________________________________ where 𝑏 > 0 and 𝑏 ≠ 1, that is the inverse of the exponential function 𝑦 = 𝑏𝑥 𝑦 = log𝑏 𝑥 where b →_____________________________ 𝑦 →_____________________________ 𝑥 →_____________________________

R7/8

MPC40S Date: ____________________

Pg. #18

Example #1

Express each of the following in logarithmic form. a) 34 = 81

b) 𝑚 = 4𝑛

c) 𝑦 = 8(𝑥+1)

d) 26 = 64

Example #2

Express each of the following in exponential form. a) log2 8 = 3

b) log7 343 = 3

c) log4 𝑥 = 3

d) log2 16 = 𝑥 + 1

MPC40S Date: ____________________

Pg. #19

Example #3

Evaluate the following expressions

a) log2 16 = 𝑥

b) log2 (1

4) c) log3(−27)

Common Logarithm: A logarithm ______________________________. When you write a common logarithm, you do not need to write the base. For example, log 3 means log10 3 Natural Logarithm: A logarithm ______________________________.

MPC40S Date: ____________________

Pg. #20

Some Basic Logs to Remember: “Quicksnappers”

a) =1log c

b) =cclog

c) =y

c clog

d) =ycclog

e) 𝑙𝑛𝑒 =

MPC40S Date: ____________________

Pg. #21

Example #4

Evaluate the following

a) log2 8

b) 4log55

c) log 1000

MPC40S Date: ____________________

Pg. #22

Example #5

Solve each of the following equations.

a) log𝑥 5 =1

2

b) log 𝑥 = −3

c) log16 𝑥 =3

4

d) log2(log3 81) = 𝑥

MPC40S Date: ____________________

Pg. #23

Example #6

Estimate the following values a) log2 30

b) log 20

MPC40S Date: ____________________

Pg. #24

MPC40S Date: ____________________

Pg. #25

Chapter 8: LOGARITHMIC FUNCTIONS

8.2 – Transformations of Logarithmic Functions Recall: The logarithmic function is the inverse of the exponential function. Remember, to find the inverse of a function, we switch the 𝑥 and 𝑦 values and then

solve for 𝑦. Exponential Function Inverse Function 𝑦 = 𝑏𝑥 𝑥 = 𝑏𝑦 which is equivalent to 𝑦 = log𝑏 𝑥 Example #1

Sketch the graphs of the functions 𝑦 = 2𝑥 and 𝑦 = log2 𝑥. Note that these two functions are inverses of each other.

Note: The graph of 𝑦 = log2 𝑥 has a _________________________________________ because ______________________________________________________________

𝑥 𝑦

𝑥 𝑦

R9

MPC40S Date: ____________________

Pg. #26

Example #2

Sketch the graphs of the following functions on the same Cartesian plane. a) 𝑦 = log3 𝑥 b) 𝑦 = log4 𝑥 c) y = log 𝑥 d) 𝑦 = ln 𝑥

Note: All the graphs pass through the point (1,0). The base of the logarithm determines the next point. Base 3 → (3,1) 𝑦 = log3 𝑥 → 3𝑦 = 𝑥 → 31 = 3 → (3, 1)

MPC40S Date: ____________________

Pg. #27

Example #3

a) Sketch the graph of the function 𝑦 = log5(𝑥 + 2) − 1 b) Sate the domain and range. Domain: __________________ Range: ___________________ c) State the equation of the asymptote. d) Determine the x – intercept.

MPC40S Date: ____________________

Pg. #28

Example #4

a) Sketch the graph of the function 𝑦 = log3(3 − 𝑥) + 1 b) Sate the domain and range. Domain: __________________ Range: ___________________ c) State the equation of the asymptote. d) Determine the y – intercept and the x – intercept.

MPC40S Date: ____________________

Pg. #29

Chapter 8: LOGARITHMIC FUNCTIONS

8.3 – Law of Logarithms 1. Example: log4 10 + log4 2 = Example: log2 6 = 2. Example: log3 15 − log3 5 =

Example: log𝑥8

3=

3.

Example: log3 √6 = Example: 3log2 5 =

log𝑎 𝑀𝑁 = log𝑎 𝑀 + log𝑎 𝑁

log𝑎

𝑀

𝑁= log𝑎 𝑀 − log𝑎 𝑁

log𝑎 𝑀𝑥 = 𝑥 log𝑎 𝑀

R7/8

MPC40S Date: ____________________

Pg. #30

4. Example: log2 7 =

The l log button on a calculator is of base 10. Therefore, this button can only be used

to find solutions with base 10. For example, log2 7 cannot be entered into the calculator. We must use the “change of base” law to convert this expression to base 10 and then a calculator can be used to solve it.

log2 7 =log 7

log 2=

Example #1

Expand following expression so that it is written in terms of individual logarithms.

log5𝑋𝑌

𝑍

Example #2

Expand the expression log𝑎 (𝐴√𝐶

𝐵2𝐷)

log𝑎 𝑥 =log 𝑥

log 𝑎

MPC40S Date: ____________________

Pg. #31

Example #3

Simplify the expression so that it is written as a single logarithm. a) log5 𝐴 + 3log5 𝐵 − log5 𝐶

b) 2 log 𝑥 − (log 𝑦 −1

3log 𝑧)

MPC40S Date: ____________________

Pg. #32

Example #4

Simplify and then evaluate the following expressions using laws of logarithms: a) log6 8 + log6 9 − log6 2

b) 2 log2 12 − (log2 6 +1

3log2 27)

MPC40S Date: ____________________

Pg. #33

Example #5

If log𝑎 2 = 0.30 and log𝑎 3 = 0.48 and log𝑎 5 = 0.70, then evaluate the following expressions: a) log𝑎 150

b) log𝑎 √10

3

MPC40S Date: ____________________

Pg. #34

MPC40S Date: ____________________

Pg. #35

Chapter 8: LOGARITHMIC FUNCTIONS

8.4 – Logarithmic and Exponential Equations We have now learned everything we need to know to solve equations! Example #1

Solve log3 𝑥 − log3 5 = −1

Example #2

Solve log6(2𝑥 − 1) = log6 11

_________________________________________ _________________________________________ _________________________________________ _________________________________________ _________________________________________ _________________________________________ _________________________________________

R10

MPC40S Date: ____________________

Pg. #36

Example #3

Solve log(𝑥 − 3) − log(𝑥 − 5) = log 5 Example #4

Solve 2log 𝑥 − log(𝑥 + 2) = log(2𝑥 − 3)

MPC40S Date: ____________________

Pg. #37

Example #5

Solve log4(11 − 𝑥) + log4(𝑥 + 6) = 2

_________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________

Example #6

Solve log6(𝑥 + 3) = 1 − log6(𝑥 + 4)

MPC40S Date: ____________________

Pg. #38

When the bases of exponents cannot be changed to the same value we must use logarithms to solve the equation. Example #6

Solve the exponential equation 3𝑥 = 8

__________________________________ __________________________________ __________________________________ __________________________________ __________________________________ __________________________________ __________________________________ __________________________________ __________________________________

Example #7

Solve 5 = 2𝑥−2. Express the answer correct to 3 decimal places.

MPC40S Date: ____________________

Pg. #39

Example #8

Solve the following equation. Express the answer correct to 3 decimal places.

19𝑥−5 = 32𝑥+1

MPC40S Date: ____________________

Pg. #40

Example #9

Solve the following equation. Express the answer correct to 3 decimal places.

2(3𝑥) = 63𝑥−1 Example #10

Solve the following equation. Express the answer correct to 3 decimal places. 𝑒𝑥−1 = 2𝑥+1

MPC40S Date: ____________________

Pg. #41

Chapter 8: LOGARITHMIC FUNCTIONS

8.4 – Applications of Logarithmic & Exponential Functions Example #1

The Richter magnitude, M, of an earthquake is defined as 𝑀 = log (𝐴

𝐴0)

where A is the amplitude of the ground motion and A0 is the amplitude associated with a “standard” earthquake. a) In 1946, in Haida Gwaii, British Columbia, an earthquake with an amplitude measuring 107.7 times A0 struck. Determine the magnitude of this earthquake on the Richter scale. b) The strongest recorded earthquake in Haida Gwaii was in 1949 and had a magnitude of 8.1 on the Richter scale. Determine how many times stronger this earthquake was than the one in 1946.

R10

MPC40S Date: ____________________

Pg. #42

Example #2

The pH scale is used to measure the acidity or alkalinity of a solution. It is defined as 𝑝𝐻 = − log(𝐻+) where H+ is the concentration of hydrogen ions measured in moles per litre (mol/L). A neutral solution, such as pure water, has a pH of 7. The closer the solution is to 0, the more acidic the solution. The closer the solution is to 14, the more alkaline the solution is. A cola drink has a pH of 2.5 whereas milk has a pH of 6.6. How many times as acidic as milk is a cola drink? This is calculated by comparing the number of ions in each substance.

MPC40S Date: ____________________

Pg. #43

Example #3

The human ear is able to detect sounds of different intensities. The intensity level, 𝛽, in

decibels, is defined as 𝛽 = 10 log𝐼

𝐼0 where 𝐼 is the intensity of the sound measured in

watts per square metre (W/m²), and 𝐼0 is 10−12 W/m² which is the threshold of hearing (the faintest sound that can be heard by a person of normal hearing). a) A truck emits a sound intensity of 0.001 W/m². Determine its decibel level. b) It is recommended a person wears protective ear gear when the intensity level is 85 dB or greater. The MTS Centre measures 110 dB when the Jets score a goal. How many times louder is the MTS Centre than the recommended maximum sound intensity?

MPC40S Date: ____________________

Pg. #44

Example #4

Compound interest is a good example of an increasing exponential function. P = initial value A = final value n = # of times the interest is compounded in one year r = interest rate, as a decimal (ex: 5% = 0.05) t = time, in years An amount of $5000 is invested at a rate of 3.1% compounded monthly. a) Determine how much interest is earned after 20 years. b) Determine how much time is needed for the initial investment to double in size.

𝐴 = 𝑃 (1 +𝑟

𝑛)

𝑛𝑡

Interest Earned = A – P

Note: Annually 1 time/year Semi-annually 2 times/year Quarterly 4 times/year Monthly 12 times/year Biweekly 26 times/year Weekly 52 times/year Daily 365 times/year

MPC40S Date: ____________________

Pg. #45

Example #5

There are 500 mice found in a field on June 1. On June 20, 800 mice are counted. If the population of mice continues to increase at the same rate, determine how many mice there will be on June 28.

Use 𝐴 = 𝑃𝑒𝑟𝑡 where P = initial value

A = final value t = time, in days r = rate of increase or decrease

Note: If r > 0, then the function increases exponentially If r < 0, then the function decreases exponentially