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Transcript of Unit 3: Logarithms (Day 1) Exponents Review Exponent Laws: 2020-04-01¢  Unit 3: Logarithms...

• 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

x x

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.

1 0 =x

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

a

a

x x

1 =− also a

a x

x =

1 also

a

aaa

x

y

x

y

y

x =

  

 =

  

 −

• Unit 3: Logarithms (Day 1)

*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:

a bb

a

xx =

If the exponent is negative, we reciprocate the base:

a b

b

a

x x

1 =

Examples

1. Evaluate:

a) 3 4

8 b) 3 2

)27(− c) 4 3

16 −

• Unit 3: Logarithms (Day 1)

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 9 x−

=

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

• Unit 3: Logarithms (Day 2)

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

• 1 2

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

• Unit 3: Logarithms (Day 3)

Defining Logarithms

Learning

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

 Convert between logarithmic and exponential functions

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) 5122 9 = ii) 2433

5 = iii) 40968 4 =

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) 1

2

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

7 5

log10

log10

log 5

x

=

=

=

log x b b x=

2

log100

log1000

log 8

10

10

2

=

= log

b x

b x=

Homework:

1. 5.2 #1-4

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

• Unit 3: Logarithms (Day 4)

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(log 2 1

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 aay x

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 loglog2log 2 1−+ CBA logloglog3

3 1 −−

DCBA log2log2log3log 2 1 −+−

• Example: Write