Accelerated Math 2 - ahsallison.weebly.com · log 125 5 8. 6 2 9. 3 1 log 9 10. 1 4 log 64. 2...
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Honors Algebra 2 Introduction to Logarithms 1 Unit 7 LOGARITHMS Lesson Opener Solve the exponential equation; show work. 1. 10 10, 000 x = 2. 10 80 x = Recall: An exponential (parent) function has the form x y b = where 0, 1 b b , also 0. y Definition of a Logarithm: log b y x = iff x b y = ; a logarithm is the inverse of an exponential. The restrictions 0, 1 b b , also 0 y apply to a logarithmic expression, function or equation. To solve the equation 10 80 x = , a logarithm answers the question “10 raised to what number equals 80?”. I can rewrite the equation 10 80 x = as 10 log 80 x = and the calculator can evaluate 10 log 80 1.9031 . Examples log b y x = x b y = Rewrite the log equation as an exponential equation. 1. 10 log 100 2 = 2. 3 log 81 4 = 3. log m n a = Rewrite the exponential equation as a log equation. 4. 5 2 32 = 5. 3 1 4 64 − = 6. d c f = Evaluate without using a calculator. 7. 5 log 125 8. 2 log 16 9. 3 1 log 9 10. 1 4 log 64
Transcript of Accelerated Math 2 - ahsallison.weebly.com · log 125 5 8. 6 2 9. 3 1 log 9 10. 1 4 log 64. 2...
Honors Algebra 2 Introduction to Logarithms
1
Unit 7 LOGARITHMS Lesson Opener
Solve the exponential equation; show work.
1. 10 10,000x = 2. 10 80x =
Recall: An exponential (parent) function has the form xy b= where 0, 1b b , also 0.y
Definition of a Logarithm: logb y x= iff xb y= ; a logarithm is the inverse of an exponential.
The restrictions 0, 1b b , also 0y apply to a logarithmic expression, function or equation.
To solve the equation 10 80x = , a logarithm answers the question “10 raised to what number equals 80?”.
I can rewrite the equation 10 80x = as 10log 80 x= and the calculator can evaluate
10log 80 1.9031 .
Examples logb y x= xb y=
Rewrite the log equation as an exponential equation.
1. 10log 100 2= 2. 3log 81 4= 3. logm n a=
Rewrite the exponential equation as a log equation.
4. 52 32= 5.
3 14
64
− = 6. dc f=
Evaluate without using a calculator.
7. 5log 125 8. 2log 16
9. 3
1log
9 10. 1
4
log 64
2
Evaluate the expression.
11. 12log 12 = 12. 47log 47 =
13. 1
2
1log
2= 14. logb b =
15. 4log 1= 16. 13log 1=
17. 2
3
log 1 18. log 1b =
Also
• 10log x is called a __________________ log and is written as____________; when you don’t see a
base we understand it means the base is 10
• loge x is called a ___________________ log and is written as ____________
• These are the logs found on a scientific or graphing calculator.
Why does the common log use base 10?
A logarithmic operation is the inverse of an exponential operation.
Simplify the expression.
19. 6log 14
6 = 20. log 310 x =
21. 7
4log 4 x = 22. 2ln xe− =
3
Log Practice #1
Do NOT use any calculator; OK to use a power table! Rewrite the equation in exponential form.
1. ln 25 3.22 2. logu v w=
Rewrite the equation in logarithmic form.
3. 53 243= 4. 10a c=
Evaluate (simplify) the following expressions without using a calculator. Work required for #7 - 8!
5. 2log 8 6. 5log5
x
7. 3
4 3log 4 log 81− 8. 2 2log 4 log 32−
9. 5 3log 25 log 3+ 10. 3 2log 27 log 16+
Solve for v.
11. 7log 343v = 12. 8
1log
3v= 13. log 625 4v =
4
x
y
Compare the graphs and characteristics of exponential and log functions.
Graph ( ) xxf 2= and its inverse ( ) xxf 2
1 log=− on the same grid.
a. Table of values for each function.
( ) xxf 2= ( ) xxf 2
1 log=−
b. Asymptote: Write equation and sketch. c. General shape of function d. Domain Range e. End behavior
( )
( )
as _______, _______
as _______, _______
x f x
x f x
→ →
→ →
f. x and y-intercepts g. Interval of increase OR decrease
b. Asymptote: Write equation and sketch. c. General shape of function d. Domain Range e. End behavior
( )
( )
as _______, _______
as _______, _______
x f x
x f x
→ →
→ →
f. x and y-intercepts g. Interval of increase OR decrease
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Honors Algebra 2 Name ____________________________ Graphing Logarithmic Functions Date __________________Block ______ How to Choose Inputs for a Parent Log Function
Identify the base (b) of the parent log and use it to find your parent table inputs. You will always get the same
outputs for the parent table using this method.
x logby x=
2b−
2log 2b b− = −
1b−
1log 1b b− = −
0b 0log 0b b =
1b 1log 1b b =
2b 2log 2b b =
Example: ( ) 5 logp x x= Final Parent Table
x 5logy x=
x 5logy x=
2 15
25
− = 2
5log 5 2− = −
1
25 2−
1 15
5
− = 1
5log 5 1− = −
1
5 1−
05 1= 0
5log 5 0=
1 0
15 5= 1
5log 5 1=
5 1
25 25= 2
5log 5 2=
25 2
6
x
y
Graph each function and provide the requested information.
1. 4logy x=
a. Table of values (parent function – 1 table) B. x-int: ___________ y-int: ___________ C. Asymptote: Write equation and sketch! Work for intercepts here if needed: D. General shape (rough sketch) E. Domain _______________Range _______________
F. End behavior ( )
( )
as _______, _______
as _______, _______
x f x
x f x
→ →
→ →
G. Intervals of increase or decrease H. State the parent function and describe the transformations in words.
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x
y 2. ( )3log 4 1y x= − +
a. 2 Tables of values (parent and transformations) B. x-int: ___________ y-int: ___________ C. Asymptote: Write equation and sketch! Work for intercepts here if needed: D. General shape (rough sketch) E. Domain _______________Range _______________
F. End behavior ( )
( )
as _______, _______
as _______, _______
x f x
x f x
→ →
→ →
G. Intervals of increase or decrease H. State the parent function and describe the transformations in words.
8
x
y 3. ( ) ( )2log 4g x x= +
a. 2 Tables of values B. x-int: ___________ y-int: ___________ C. Asymptote: Write equation and sketch! Work for intercepts here if needed: D. General shape (rough sketch) E. Domain _______________Range _______________
F. End behavior ( )
( )
as _______, _______
as _______, _______
x f x
x f x
→ →
→ →
G. Intervals of increase or decrease H. State the parent function and describe the transformations in words.
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x
y 4. ( ) 3log 1h x x= − +
a. 2 Tables of values B. x-int: ___________ y-int: ___________ C. Asymptote: Write equation and sketch! Work for intercepts here if needed: D. General shape (rough sketch) E. Domain _______________Range _______________
F. End behavior ( )
( )
as _______, _______
as _______, _______
x f x
x f x
→ →
→ →
G. Intervals of increase or decrease H. State the parent function and describe the transformations in words.
10
x
y
5. ( ) ( )32log 1 4j x x= + −
A. 2 Tables of values B. All intercepts (x and y) C. Asymptote: Write equation and sketch! Work for intercepts here if needed: D. General shape (rough sketch) E. Domain _______________ Range _______________
F. End behavior ( )
( )
as _______, _______
as _______, _______
x f x
x f x
→ →
→ →
G. Intervals of increase or decrease H. State the parent function and describe the transformations in words.
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x
y 6. ( )34log 3 1y x= + +
A. 2 Tables of values B. All intercepts (x and y) C. Asymptote: Write equation and sketch! Work for intercepts here if needed: D. General shape (rough sketch) E. Domain _______________ Range _______________
F. End behavior ( )
( )
as _______, _______
as _______, _______
x f x
x f x
→ →
→ →
G. Intervals of increase or decrease H. State the parent function and describe the transformations in words.
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x
y 7. ( )23log 1 6y x= + +
A. 2 Tables of values B. All intercepts (x and y) C. Asymptote: Write equation and sketch! Work for intercepts here if needed: D. General shape (rough sketch) E. Domain _______________ Range _______________
F. End behavior ( )
( )
as _______, _______
as _______, _______
x f x
x f x
→ →
→ →
G. Intervals of increase or decrease H. State the parent function and describe the transformations in words.
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x
y 8. ( )22log 8 4y x= − + +
A. 2 Tables of values B. All intercepts (x and y) C. Asymptote: Write equation and sketch! Work for intercepts here if needed: D. General shape (rough sketch) E. Domain _______________ Range _______________
F. End behavior ( )
( )
as _______, _______
as _______, _______
x f x
x f x
→ →
→ →
G. Intervals of increase or decrease H. State the parent function and describe the transformations in words.
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x
y
9. ( )42log 3 1y x= − + −
A. 2 Tables of values B. All intercepts (x and y) C. Asymptote: Write equation and sketch! Work for intercepts here if needed: D. General shape (rough sketch) E. Domain _______________ Range _______________
F. End behavior ( )
( )
as _______, _______
as _______, _______
x f x
x f x
→ →
→ →
G. Intervals of increase or decrease H. State the parent function and describe the transformations in words.
15
x
y
10. ( )4
1log 2 3
2y x= − +
A. 2 Tables of values B. All intercepts (x and y) C. Asymptote: Write equation and sketch! Work for intercepts here if needed: D. General shape (rough sketch) E. Domain _______________ Range _______________
F. End behavior ( )
( )
as _______, _______
as _______, _______
x f x
x f x
→ →
→ →
G. Intervals of increase or decrease
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H. State the parent function and describe the transformations in words. Honors Algebra 2 Summary of Function Families in Algebra 2
Name
General Form (h,k) Example Parent Function a, h, and k
Polynomial Unit 3
( ) ( )n
f x a x h k= − +
Parent: ( ) np x x=
( )2
3 5 4y x= − +
Quadratic ( ) 2
2 (degree)
so
n
p x x
=
=
3
5
4
a
h
k
=
=
=
Polynomial Unit 3
( ) ( )n
f x a x h k= − +
Parent: ( ) np x x=
( )3
7 2y x= − + −
Cubic ( ) 3
3 (degree)
so
n
p x x
=
=
1
7
2
a
h
k
= −
= −
= −
Radical Unit 4
( ) nf x a x h k= − +
Parent: ( ) np x x=
4 3y x= + +
Square Root ( )
2 (index or root)
so
n
p x x
=
=
1
4
3
a
h
k
=
= −
=
Radical Unit 4
( ) nf x a x h k= − +
Parent: ( ) np x x=
( ) 315
4f x x= −
Cube Root ( ) 3
3 (index or root)
so
n
p x x
=
=
1
4
0
5
a
h
k
=
=
= −
Exponential Unit 5
( ) ( )x h
f x a b k−
= +
Parent: ( ) xp x b=
( )1
4 2 8x
y−
= +
Growth ( )
2 (base)
so 2x
b
p x
=
=
4
1
8
a
h
k
=
=
=
Exponential Unit 5
( ) ( )x h
f x a b k−
= +
Parent: ( ) xp x b=
23 1
2 3
x
y
+
=
Decay ( )
1(base)
3
1so
3
x
b
p x
=
=
3
2
2
0
a
h
k
=
= −
=
Logarithm Unit 6
( ) ( )logbf x a x h k= − +
Parent: ( ) logbp x x=
( )46log 3 2y x= − + −
( ) 4
4 (base)
so log
b
p x x
=
=
6
3
2
a
h
k
= −
= −
= −
Review of transformations for all functions based on a, h, and k.
If 0a , the parent graph has been vertically reflected or reflected across the x-axis.
If 1a , the parent graph has been vertically stretched by a factor of |a|.
If 0 1a , the parent graph has been vertically compressed by a factor of |a|.
If 0h , the parent graph has shifted left h units, if 0h , the parent graph has shifted right h units.
If 0k , the parent graph has shifted down k units, if 0k , the parent graph has shifted up k units.
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Also remember:
• When expanding a log expression, rewrite radicals using rational exponents and apply
power property.
• When condensing a log expression, rewrite rational exponents as radicals (simplify the
radical if possible).
• Always look for opportunities to simplify expressions as much as possible.
Expand and simplify the following logarithm expressions:
1. 2
3
7log
x
y 2.
3
2
5log
4
y
x
3. 2
2log
2
x 4.
5 2
log7
x y z
x
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Condense and simplify the following logarithm expressions:
5. 4 4 4log 2 3log 3 log 9+ − 6. ln3 2ln lnx y+ −
7. − +3 3 3log log 2 2log 4x 8. + −
14log log 3log2
2m n p
Change of base formula: log
loglog
b
mm
b= or
lnlog
lnb
mm
b=
Rewrite and evaluate the expression, round answers to the nearest hundredth.
9. 7log 4 10. 3log 8 4−
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Complete all problems in Sections I and II OR Sections II and III. Copy problems and show work
on a separate sheet of paper.
Section I
Section II
Section III Expand and simplify the following logarithm expressions:
1. 2
8log 3x y 2. 2
3log 9x y
3. 4
5ln
8
e
x
4. 238log 64 xy
Condense and simplify the following logarithm expressions:
5. log log log4 4 46 2 5+ + 6. log 4log 7 log 49b b bx+ −
7. ( )1
log 4 log 3log 2 4log2
x z− + − 8. 3ln (4ln ln )x y z− +
Given: 2log 3a = 2log 3b = −
2log 2c = 2log 4d = −
Find:
9. 2log
ab
cd
= ____________ 10. 3 2
2log a d = ___________
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Solutions to Sections I and II
Solutions to Section III
1. 8 8 8log 3 2log logx y+ +
3. 4 ln8 5ln− − x
5. 4log 60
7. 4
16log
z x
9. 2log
ab
cd
= 2
2. 3 32 2log log+ +x y
4. 8 8
1 22 log log
3 3x y+ +
6. log 49b x
8. 3
4ln
x
y z
10. 3 2
2log a d = 1
2log 2 x - 2
