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

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  • 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

    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) 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