Download - Unit 3: Logarithms (Day 1) Exponents Review Exponent Laws: … · 2020-04-01 · Unit 3: Logarithms (Day 1) Exponents Review Exponent Laws: Multiplication Law: When multiplying powers

Unit 3: Logarithms (Day 1)

Exponents Review

Exponent Laws:

Multiplication Law:

When multiplying powers with the same base we add the exponents. baba xxx +=

ex: = 65 xx

Division Law:

When dividing powers with the same base we subtract the exponents.

baba xxx −= or ba

b

a

xx

x −=

ex. = 310 xx ex. 6

2

x

x

Power Law:

When a power is raised to an exponent we multiply the exponents. baba xx =)(

ex. 73)(x

* The power law is always done before the multiplication and division laws *

iv) 322 )2)(5( baab − v) 22

3422

)4(

)8)(6(

mn

nmnm

−

Zero Law: Any base (except 0) to the power of zero equals 1.

10 =x

Negative Law: A negative exponent is changed to a positive by reciprocating the base.

a

a

xx

1=− also a

ax

x=

−

1 also

a

aaa

x

y

x

y

y

x=

=

−


*We never leave negative exponents in our answer

Examples: Evaluate:

i) 23− ii)

2

5

4−

iii) 42

1−

iv) 1)75.0( −

v)

3

3

2−

− vi) 2)4.0( −

ex: Rewrite with positive exponents:

i) 263

122−−

−−

fed

bca ii)

31

2343

ts

rqp−

−−−

Fractional Exponents: can be converted to radicals by the following rule:

abb

a

xx =

If the exponent is negative, we reciprocate the base:

ab

b

a

xx

1=

−

Examples

1. Evaluate:

a) 3

4

8 b) 3

2

)27(− c) 4

3

16−


Solving Exponential Equations with the Same Bases

Re-write the following as a power of a different base:

i) 9x

ii) 2125 x−

iii) 8 16x

We can use the above strategy for solving exponential equations:

Example: Solve

i) 1 22 8x x+ =

ii)

4 1

1 14

2

x

x

−

− =

iii) ( )1

33 9x−

=

iv)

1

2 3125 125

5

x

x

−

+ =

What About 12 5x x+ = ????

Homework:

WS 2-03 Exponent Review

5.1 # 1, 2

WS 2-18 Solving Equations Using Properties of Exponentials


Graphing and Analyzing the Exponential Function

Exponential Functions are of the form: xy a=

Sketch: ( ) 2xf x = and ( ) 5xg x =

Asymptote: Features: Asymptote: Features:

Table of Values: x y x y

12

x

y

=

Asymptote:

Features:

Generally, for the exponential function:

xy a= , 1a Asymptote: 0y = xy a= , 0 1a Asymptote: 0y =

Features: Features:

Special Cases:

What if 0? 0 Undefined for 0xa y x= = 1, 1 1xIf a y y= = =

What does ( 3)xy = − look like (e.g., 0a )

Sketch the following:

1. 10 1xy = +

2. 22xy −=

3. 3 xy −=

4.

5. 63 5xy −+ =

Homework

1. 5.1 #3-6, 8

2. Analyzing and Graphing Exponentials WS


Defining Logarithms

Learning

Intention(s): Investigate logarithmic functions and relate them to exponential functions

Convert between logarithmic and exponential functions

Use your calculator to evaluate

log10

log100

log1000

log10000

log x is the exponent 10 has if x is written as a power of ten.

Definition:

“log base a of x equals y”:

The answer, y, is the exponent a has if written as a power.

** xx 10loglog =

If there is no number in the base position, the base is 10

Examples: Convert to exponential form:

i) 3216log6 = ii) 664log 2 = iii) 01log =

Examples: Convert to logarithmic form:

i) 51229 = ii) 24335 = iii) 409684 =

To evaluate simple logarithms, convert to exponential form.

Examples: Evaluate:

i) 8log 2 = ii) 81log3 =

iii) 125log5 = iv) 49log7 =

v) log 10,000,000,000 =

vi) 125log25 vii) )82(log4

Write 2 as a logarithm with base 7

Estimate 18log 2

We can evaluate logs with base 10 on our calculators.

Examples: Evaluate to 4 decimal places

i) log 20 ii) log 120

iii) log 1200

_________________________________________________________

Ex: Find the inverse of:

i) xy 2= is iii) xy 3= is iii) xy 10= is

Sometimes, after converting to exponential form you may need to

re-write using a common base to solve.

Examples:

Evaluate

i) 4log 8 ii)

12

log 4

iii) 27 9log iv) 4log (2 8)

Evaluate the following:

( )log 2− ( )log 0 2log 8− 2(log 8)− − 1log 15

What can you conclude from the examples above?

Some Simplification Rules (Tricks )

2

75

log10

log10

log 5

x

=

=

=

log xbb x=

2

log100

log1000

log 8

10

10

2

=

= log

bx

b x=

Homework:

1. 5.2 #1-4

2. Extra Practice: WS 2-5 to 2-6


Graphing Logarithmic Functions

Consider: xy b= → →

***A logarithm is the inverse of an exponential.

Ex: Find the inverse of each of the following:

1. Switch x and y

2. Isolate the exponential/logarithm

3. Change to a logarithm/exponential

i) 15 3 −= +xy iii) 3)1(log8 ++= xy

__________________________________________________________________________________

i) Make a table of values and sketch a graph of xy 2log=

Properties of the graph:

x-intercept:

y-intercept:

Domain:

Range:

Asymptotes:

In general: logay x= , if 1a and logay x= , if 0 1a

******************

Comparing Features of Exponential and Logarithmic Graphs

xy a= logay x=

Transforming the graph: qpxbay +−= )(log

• a expands the graph vertically

• b expands the graph horizontally (as the reciprocal)

• p translates the graph left/right

• q translates the graph up/down

Examples: Describe how the following compare to xy 2log= . Then determine the domain.

4)3(log2 −−= xy 1)3(2log2 −+= xy

2)1(log2 2 ++= xy 2)2(log21

2 −−= xy

Example:

Determine the domain: 1log ( 3)xy x−= +

How to Graph Log Functions:

i) Determine parent function

ii) Determine and plot the key points of the parent function

iii) Follow the indicated transformations

iv) Graph the result

Graph:

2log ( 3) 4y x= + − 1

2

2log ( 1) 2y x= + +

Assignment:

1. 5.2 #5-13

2. WS Logs Lesson 4 Assignment

Unit 3: Exponents and Logarithms (Day 5)

Laws of Logarithms

Learning

Intention(s): Apply logarithm laws to simplify and solve equations

Exponents Logarithms

Multiplication Law: yxyx aaa += yxxy aaa logloglog +=

Division Law: yxyx aaa −= yx aayx

a loglog)(log −=

Power Law: xnnx aa =)( xnx a

n

a loglog =

Using the Laws:

Example: Simplify

i) 4log25log + ii) 640log64log −

iii) 5.1log6log 33 + iv) 2log3log4log24log 8888 +−+

Write as a single logarithm:

CBA loglog2log21−+ CBA logloglog3

31 −−

DCBA log2log2log3log21 −+−

Example: Write in terms of log a, log b and log c are related?

i)

2

3

logc

ba ii)

53

2

logcb

a

Example: If 6log =x and 2log =y and 3log =z , simplify

yz

x2

log

Example:

If log 5 = a and log 2 = b, write the following in terms of a and b.

log 200

log 80

Write as a single log:

24log3log 22 ++ 6log32log 33 −+

9log2

16log2log3 444 −+

)23log()4log( 22 ++−− xxx

Homework:

5.3 #1-5


More Laws of Logarithms

Helpful Rules!

loglog

log

xb

x

aa

b= loga ba b= b>0

1log

logb

a

ab

= log log

log log

a b

a b

x x

y y=

Write the following in terms of log3 and log 4

i) log12 ii) log 36

Find the exact value

i) 1

47log 7

ii) 6 2

5 5log 5 log 5−

Using The Change of Base Rule. Change the base to simplify expressions with different bases:

loglog

log

xb

x

aa

b=

Examples:

1. Evaluate to 5 decimal places

i) 37log 4 ii) 80log 2

2. Evaluate 25log 125

3. Simplify 9 276log 12logx x−

More Change of Base Rule:

12log

37log

4

4 12log

37log

15

15

Why does loga bba = ?

Simplify:

i) log 3 log 5a aa +

ii) 2log 6 log 12a aa −

Simplify:

2 5

1 1

log 10 log 10+

Homework:

5.3 #1-5


Solving Exponential Equations

We are going to solve an exponential equation in 4 different ways. Example: 1 22 8x x+ =

1) Re-write the bases and equate the exponents to solve for x

2) Take the base 10 log of both sides (Use Power Law)

3) Take the NOT BASE 10 log of both sides (Use Power Law)

4) Use a graphing calculator

So if given 2x+1 = 72x, which methods could you apply?

In Summary, if possible, convert to the same base and solve. Otherwise, log both sides and solve

using log laws:

1. Isolate the exponential

2. Take the log of both sides

3. Use the power law for logarithms

4. Solve for x

Examples: Solve in terms of logarithms and then give answer to 4 decimal places.

i) 203 =x ii) 480012.1120 = n

iii) 352 53 =−x

iv) ( ) 12 6 12x x+=

v) 3 (2)x = 12

Homework: WS 2-17 (omit #8,10) and Workbook p.227 #4


Solving Logarithmic Equations

To solve a log equation, manipulate the equation to fit one of two situations:

Type I Type 2

log (…….) = log (….…)

Strategy: Cancel out the logs and solve for x

log (…....) = …….

Strategy: Change to exponential form and

solve for x

Examples: Solve

i) 5 5 5log log ( 3) log 10x x+ − =


ii) ( ) ( )8 8log 2 1 log 4x x− = − − ii) 2log log 64x =

iii) 34log 12x =

Homework: WS 2-19 to 2-22 (omit questions with ln x ) and p.227#1-3,5,7,8


Applications of Exponentials and Logarithmic Functions DAY 1

Exponential Equations are used to model:

- Compound Interest – eg. Investments

- Exponential Growth – eg. Cell division

- Exponential Decay – eg. Half-life

For all of these problems we use the same formula:

(1 )t

TP A r= +

P = Final Amount

A = Initial Amount

r = rate of growth/decay (% - in decimal form)

t = actual time passed

T = length of one time period

Examples of Exponential Growth:

1. You invest $2000 at 5% interest compounded annually.

a) How much money do you have after 5 years?

b) How long does it take to double your investment?

2. Your company invests $50,000 at 6% annual interest compounded quarterly. How long does it

take for this investment to double?

3. You invest $5000 at 6% annual interest compounded monthly. How long does it take for your

investment to double?

4. Alberta’s population (in 1981) was 2.28 million and was growing at a rate of 1.4% per year. How

long would it take to reach 4 million? (Assuming the rate of growth remains constant)

5. Canada’s population (in 1985) was 29.6 million and was growing at a rate of 1.24% per year.

Estimate the doubling time for Canada’s population.

Examples of Exponential Decay:

6. After drinking coffee, the percentage of caffeine in a person’s body decreases by 13% per hour.

a) How long does it take for there to be 50% of the caffeine left?

b) Write the equation for caffeine dissipation as an exponential with base 0.5 (as a half life)

7. Radioactive iodine-131 (from the Chernobyl explosion) has a half-life of 8.1 days. How long did it

take to reduce to 1% of the level immediately after explosion?


Applications of Exponential and Logarithmic Functions DAY 2

Exponential Growth can always be modeled by: T

t

rAP )1( +=

Suppose you make $10 an hour and you get a 100% raise. How much will you now make per hour?

What if you get a 200% raise?

A 900% raise?

Doubling: an increase of 100%.

Tripling: an increase of 200%

Increasing tenfold: an increase of 900%

**************************

1. A bacteria doubles in size every 2 hours. Write the equation for the size of the culture after t hours if

the culture had only 1 cell to start.

2. A population of rabbits triples every 4 months. Write the equation for the population size after t

months if there are 2 rabbits to start.

3. The population of a swarm of locusts multiplies tenfold every 20 days.

a) If there are currently 2 million locusts, how many are there after 1 year?

b) How many days does it take for the swarm to double?

c) How many times as great is the swarm after 12 days than after 3 days?

The Richter Scale

Richter scale magnitudes are based on a logarithmic scale (base 10). What this means is that for each

whole number you go up on the Richter scale, the amplitude of the ground motion recorded by a

seismograph goes up ten times.

Increase of 1 whole unit on the scale corresponds to a ten-fold increase in intensity:

6.0 earthquake is 10 times as intense as 5.0

6.0 earthquake is 10 x 10 times as intense as 4.0

6.0 earthquake is 10 x 10 x 10 times as intense as 3.0

4. The 1989 San Francisco earthquake measured 6.9. The 1964 Alaska earthquake measured 8.5. The

1985 Mexico City earthquake measured 8.1.

a) What is the magnitude on the Richter scale of an earthquake that is twice as intense as the Mexico

City earthquake?

b) How many times intense as the San Francisco earthquake was the Alaska earthquake? The Mexico

City earthquake?

The pH Scale

The pH scale is logarithmic and as a result, each whole pH value below 7 is ten times more acidic than

the next higher value.

• Each increase of 1 above 7 is a 10 fold increase in alkalinity

• Each decrease of 1 below 7 is a 10 time increase in acidity

5. The pH of your drinking water was tested at 7.2. How many times more alkaline is this than pure

water?

6. What is the pH of a substance that is 200 times more acidic than orange juice?

The Decibel Scale

The loudness of sound is measured in decibels (dB). Every increase of 10 dB is a tenfold increase in

loudness.

60 80 100

Conversation (60 dB) Average Traffic (85dB) Chain Saw (100dB)

a) How many times louder is traffic than a conversation?

b) How many times louder is a chain saw than traffic?

c) What is the measure in dB of a sound that is twice as a loud as a chainsaw?

Unit 3: Exponents and Logarithms (Review)

Unit Review

Topics Covered:

1. Definition of a logarithm: xayx y

a ==log , 1,0,0 xxa

2. Logarithm Laws:

• Multiplication Law: yxxy aaa logloglog +

• Division Law: yx aayx

a loglog)(log −

• Power Law: xyx a

y

a loglog

• Change of Base: a

bba

log

loglog

3. Exponential Equations

• If possible, convert to the same base and solve

• Otherwise, log both sides and solve using log laws

4. Logarithmic Equations

5. Graphing Exponential Functions: qbaseay pxb += − )()(

6. Graphing Logarithmic Functions: qpxbay base +−= )]([log

7. Exponential Growth and Decay: T

t

rAP )1( +=

• Population growth

• Compound Interest

• Half-Life

• Earthquakes/pH/Decibels

Examples:

1. Simplify:

a) 3log218log2log +− b) rqp aaa log4log3log3

1−−

2. x and y are related as follows. How are log x and log y related?

y = 310x

3. Solve the following:

a) 543 625 +− = xx

b) 312 8127 +− = xx

c) )2(log1)4(log 77 −−=+ xx

d) 2))43((loglog 23 =+x

4. Simplify:

a) 4log5log aaa

+ b)

2

loglog

+

b

ab aa

c) 10

32

8

4 loglog xx +

5. Graph the following. State the Domain and Range for each.

a) 53 3 −= −−xy b) )4(log3 2 +−=− xy

6. If 10 b , which is larger: 4logb or 8logb ?

7. You invest $4,000 at 6% compounded monthly. How much will your investment be

worth after 3 years?

8. If is estimated that 40% of a radioactive substance decays in 24 hours. What is the

half-life of this substance?

9. Earthquake data: Columbia 1983 (5.5), Peru 1970 (7.7), BC coast 1700 (9.0)

Each 1 point increase on the Richter scale is a 10-fold increase in intensity. What is

the Richter scale reading of an earthquake that is 20 times as intense as the Columbia

earthquake?

10. A swarm of locusts grew from 100 to 21 000 in 7 days. What is the doubling time?