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PAP Algebra II Notes 7.5 Solving Exponents Same Base To solve exponential equations, get the same base on both sides of the = sign. Then the exponents must equal each other for the two sides to be equal.
EX1) 164 x EX2)
125
15 x EX3)
32
124 x
EX4) 813 25 x EX5)
36
16 x
1
PAP Algebra II WS 7.5 NONCALCULATOR Name: _________________ Solve the following exponential equations for x. SHOW ALL WORK!
1) 1255 x 2) 273 x 3)
64
14 12 x
4) 49
173 36 x
5) 12525
5 1
x
x
6) 412
27
19 x
7) x
x
32
143
8) 625
15 23 x
9) 12 279 xx
3
7.6 Understanding Logarithms WS Name_________________________ Rewrite each equation in logarithmic form. 1. 34 = 81 2. 26 = 64 3. 53 = 125 4. 80 = 1
5. 4-2 = 16
1 6. 3-1 =
3
1 7. 2-4 =
16
1 8. 7-2 =
49
1
9. 3 21
= 3 10. 9 23
= 27 11. 36 23
= 216 12.
9
1 2 = 81
Rewrite each equation in exponential form. 13. log 2 32 = 5 14. log 8 64 = 2 15. log 11121 = 2
16. log 13 13 = 1 17. log 5 1 = 0 18. log 3 243 = 5
19. log2
1 16 = -4 20. log 8 4 = 3
2 21. log 10
10
1 = -1
22. log 525
1 = -2 23. log
3
1 81 = -4 24. log 27 3 = 3
1
Evaluate each expression 25. log 10 1000 26. log 6 36 27. log 12 144 28. log 10 0.01
29. log
4
1 64 30. log 4 2 31. log 9 27 32. log 8 16
33. log
2
1 8 34. log 10 0.001
5
Solve each equation. 35. log b 49 = 2 36. log b 64 = 3
37. log 6 x = 2 38. log 9 x = -1
39. log
2
1 16 = x 40. log 3 27 = x
41. log b 81 = 4 42. log b 18 = 1
43. log 5 x = -2 44. log 3 x = -3
45. log 10 10 = x 46. log 5 5 = x
47. log a27
1= -3 48. log b 36 = -2
49. log2
1 x = -6 50. log 4 x = -2
1
51. log 2 x = -4 52. log
3x = 6
53. log3
27 = x 54. log x 5 = 4
1
55. log x3 7 =
3
1 56. log 10
3 10 = x
57. logx
6 = 2 58. log2
116
1 = x2
6
PAP Algebra II
7.7A Graphing Logs & Log Properties Notes
acb log cba
EX1: Write each log equation in exponential form.
a) 225log5 b) 4log81
13 c) 1)1log(. d) 30855.20ln
EX2: Find the value of each log WITHOUT A CALCULATOR.
a) 16log2 b) 81
13log c) )001log(.
d) )7(log 10
7 e) 3ln e f)
564
14log
The same transformations A, B, C and D that we used for other functions work the same way for exponential functions.
DCxBAy b ))((log
In order to graph xy 2log we first need to graph xy 2 and use the properties of
inverses:
1) If the point (x,y) is on f(x), then (y,x) is on the inverse of f(x).
2) The graph of a function and it’s inverse are reflected across
the line y=x.
7
xy 2 xy 2log
EX3: Graph the following exponential functions without a calculator. Label at least 2
points and any asymptotes.
a) )1(log3 2 xy b) 2log3
12 xy
EX4: Graph…
a) xy 5log b) 5log 3 y x
8
THREE LOG PROPERTIES:
1. )(logloglog MNNM bbb addition / multiplication
2. NM
bbb NM logloglog subtraction / divison
3. MKM b
K
b loglog exponent
EX5: Use these log properties to expand the following logs completely.
a) 2
7log b a
b) 45 2ln
y
x
c) 2
3log 4b
d) 2
3ln
x
y
9
EX6: Condense the following into a single log. Write answers in radical form.
a) srq 6621
6 log4loglog5
b) zxw logloglog251
31
c) )lnln5)(ln(3
1
2
1 dcba
EX7: If 35log a and 108log a find…
a) 40log a b) 8log a c) 25log a
10
PAP Algebra II
WS 7.7A NONCALCULATOR Name: _______________________
Write each equation in exponential form.
________ 1) 216log4 = ________ 2) ( ) 2log36
16 −= ________ 3) 31000log =
________ 4) 1ln =e ________ 5) 481log3 = ________ 6) 01ln =
Find the value of each log.
________ 7) 32log2 ________ 8) 9log3 ________ 9) 100log
________ 10) ( )16
12log ________ 11) ( )
125
15log ________ 12) 01.log
________ 13) ( )10
2 2log ________ 14) ( )7
6 6log ________ 15) ( )910log
________ 16) eln ________ 17) 1ln ________ 18) 4lne
________ 19) eln ________ 20) ( )e
1ln ________ 21) 45 5log
________ 22) 36
16log ________ 23) 3log9 ________ 24) 7 1.log
Graph the following exponential functions. Label at least 2 points and any asymptotes.
Write domain and range for each of the functions.
25) xy 3log= 26) )2(log4 3 −= xy
11
27) 4log2 3 +−= xy 28) 3log 2= −y x
29) xy 4log= 30) ( ))3(log21
4 += xy
31) xy ln= hint: 718.2≈e 32) 3ln2 +−= xy
12
33) If 3( ) log ( 4)= −f x x , find 1( )−f x .
34) If 3( ) 2 4−= +
xg x , find 1( )−g x .
Expand completely.
____________________________ 35) ( )3ln MN
____________________________ 36) ( )37log
y
x
___________________________ 37) 3
25
2
logC
D
__________________________ 38) ( )P
MLlog
___________________________39) ( )23
3log ba +
___________________________ 40) 41
ln−t
rs
13
Condense into a single log. Write fraction exponents in radical form.
41) RQ 32
13 loglog2 + 42) ( )zyx ln3lnln
2
1 −+
43) ( )fed 995
193
2 log2loglog +− 44) ( ) ( )srqp ln2ln7lnln53
1 +−−
45) cba 223
12 logloglog3 +− 46) )log)(log(4 fgf −+
47) Given 72log =b and 126log =b find…
a) 12logb b) 3logb c) 36logb d) 3 6logb e) 8logb
14
7.7B Properties of Logs Notes Pre-AP Algebra 2
#1 Product Property: log b MN = log b M + log b N
#2 Quotient Property: log bN
M = log b M - log b N
#3 Power Property: log b MK = K log b M
For 1-5, write each expression as a single log. 1. log 7 + log 2 2. log 5 + log 8 – 2 log 2 3. 4 log m – log n 4. log 8 – 2 log 2 + log 3 5. 3 log 2 + log 4 – log 16
For 6 – 10, expand.
6. log 5s
r 7. log 3 7(2x – 3)2 8. log 3m4n-2
15
9. log 4 (4mn)5 10. logy
x2
For 11 and 12, evaluate.
11. log 5 5 - log 5 125 12. 5log 3 3 - log3
9
13. Expand log 20 in three different ways.
16
7.7B Properties of Logarithms Worksheet Name _____________________ Express as a sum or difference of logarithms and evaluate, if possible. 1. log 3 (9 81) __________________________ 2. log 2 (4 8) ____________________________
3. log b 7y _____________________________ 4. log q Dx ______________________________
5. log a2
45 _____________________________ 6. log b
5
B ______________________________
Express as a product. 7. log a x4 ______________ 8. log e t6 _________________ 9. log b y3x _____________ 10. log k 25 ___________
Express as a single logarithm and simplify, if possible. 11. log 2x + log (3x – 7) 12. log a x3 + 3 log a y
_________________________________________ __________________________________________
13. 3
1log 2 x – 2 log 2 y + 2 log 2 x 14. log a (x2 - 9) - log a (x + 3)
_________________________________________ __________________________________________
15. log a 2x + 2 log a x 16. 2 log k + 3 log m – log (n + 10)
_________________________________________ __________________________________________ Expand. 17. log a 4xy2z4 _________________________________________________________________
18. log a 3
2
z
yx __________________________________________________________________
17
19. log a 2yz
x _________________________________________________________________
20. log b 7x3yz2 __________________________________________________________________
21. log 5 252y
x_________________________________________________________________
22. log 3 3
23
z
yx __________________________________________________________________
Given log12 2 = 0.301 and log12 5 = 0.699, find the following without a calculator.
23. log12 25 ____________________ 24. log122
1 _______________________
25. log125
2___________________ 26. log12 20 _______________________
27. log123 5 ____________________ 28. log12 8 ________________________
18
Notes 7.8: Transformations of Logs
PAP Algebra 2 The General Form of a Logarithmic (Log) Equation is
y = a log b (x - h) + k, where when b is not listed, it is 10.
Describe the effects each of the following have on the parent function?
h:
k:
a:
First graph the parent function, then…
Graph y = log (x + 1) – 3 Graph y = -2 log (x) + 4
Domain _____________________ Domain ___________________
Range ______________________ Range _____________________
Vertical Asymptote ____________ Vertical Asymptote ____________
19
If f x x( ) log 10, describe the following transformations.
a. g x x( ) log 3 510 b. g x x( ) log ( ) 10 2
c. g x x( ) log ( ) 1
22 210
Write an equation g(x) for the function ( ) logf x x with the following transformations.
A. Translated right 3 units, up 5 units, and vertically stretched by a factor of 4.
B. Reflected over the x-axis, translated left 2 units, down 3 units, and vertically compressed by a factor of 1/3.
The function f(x) = 3logx + 2 is transformed so that the new function g(x) has the following points. Describe the transformation from f(x) to g(x) and write the equation of g(x). a. b. c.
x y
2 3
4 3.9031
6 4.4314
8 4.8062
x y
-2 -2
-1 -2.9031
0 -3.4314
1 -3.8062
x y
-1 0
-2 0.9031
-3 1.4314
-4 1.8062
20
Worksheet 7.8 Name: _____________________________Per_______ PAP Algebra 2 Write each equation in logarithmic form.
1. 2 32x 2. 2 14
16
Write each equation in exponential form.
3. 8
2log 4
3 4. log(0.001) 3
Verbally describe the transformations of the parent function for the following:
5. a) y = 4 log (x – 1) – 3
b) y = -3
4 log (x) + 10
c) y = 1
6 log (x + 2) – 4
6. Graph the following using your knowledge of transformations. Then state the Domain and Range. a) f(x) = -3log(x) + 2 Transformation(s):____________ __________________________ Domain: ____________________ Range: _____________________
b) f(x) = 1
3log(x + 2) – 1
Transformation(s):____________ __________________________ Domain: ____________________ Range: _____________________ 7. Write an equation for the graph of the parent function y = log x given the following transformations:
a) Vertical compression by a scale factor of two-fifths. Horizontal translation right 8 units and a vertical translation down 4 units.
b) Reflected over the x-axis and vertically stretched by a scale factor of 6. Horizontal translation left 3 units and a vertical translation up 2 units.
c) Horizontal translation right 7 units and vertical translation up 1 unit. 21
8. Graph f(x) and g(x) on the same coordinate plane:
4logy x
4log ( 4) 1y x
Domain: __________________ Domain: __________________ Range: ___________________ Range: ___________________ 9. The function f(x) = 2logx + 1 is transformed so that the new function g(x) has the following points. Describe the transformation from f(x) to g(x) and write the equation of g(x).
a. b. c.
10. Which of the following is the inverse of 3 5 xy ?
A. 3log 5 y x B. 3log 5 y x C. 3log 5 y x D. 3log 5y x
11. Which of the following is the inverse of 5log ( 2) y x ?
A. 5 2 xy B. 5 2 xy C. 25 xy D. 25 xy
12. Condense: 4log 3(log 2log ) x z y
13: Expand: 2
3log6
x y m
14. If a and b are positive integers and 3 81a b , what is a
b?
X Y
¼
1
4
16
X Y
x y 2 2
4 2.9542 6 3.3979
8 3.6902
x y
1 1.6021
3 2.5563
5 3
7 3.2923
x y
-2 2 -1 2.6021
0 2.9542 1 3.2041
22
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23
Notes 7.9 Applications for Logarithms Name _____________________________ Richter Scale of Earthquake Intensity The relationship between the magnitude of an earthquake and its intensity is logarithmic. Furthermore, the magnitude of an earthquake is related to the logarithm of the ratio of the intensity of the earthquake to the
intensity of an earthquake of minimal intensity. If 0I =1 is the intensity of an earthquake that can just be felt and I
is the intensity of the earthquake being measured, the magnitude ( )R of the earthquake on the Richter scale is the
function 0
logI
RI
.
1. The largest recorded earthquake in the United States was a magnitude 9.2 that struck Prince William Sound, Alaska on Good Friday, March 28, 1964 UTC. What was its intensity?
2. The largest recorded earthquake in the world was a magnitude 9.5 (Mw) in Chile on May 22, 1960. What was its intensity?
3. The most recent earthquake in Chile was on Saturday Feb 27, 2010 and had a magnitude of 8.8. What was its intensity?
4. How much stronger was the Chile earthquake of magnitude 9.5 then the one with magnitude of 8.8.
5. What is the magnitude of an earthquake that is 4000 times as intense as 0I ? That is, 04000I I .
6. Matt E. Matics invests $10,000 at 8% annual interest and leaves it there for 10 years. What is the total amount in his account at the end of the 10 years?
7. Stuart Dent invests $1000 in an account earning 7.3% compounded continuously. How many years will it
take for Stu’s money to double?
24
8. Rita Chapter invests $10,000 at 8% annual interest compounded monthly. To how much will her money
grow in 10 years? Use nt
nr
PA
1 , where r = the annual interest rate, n = the number of compounding
periods per year, t = the time in years, P = the principal, and A = the amount in the account after t years. 9. Anita Lone invests $100,000 at 8.3% annual interest compounded quarterly. After how many years will she
have $150,000 in her account? Use nt
nr
PA
1 ?
10. Al Jebra buys a car for $19,500. The average annual depreciation on this model is 13%. What will the car
be worth in four years when Al finally pays off his loan? 11. From question #5 above, when will the value of the car be half its current value? 12. The population of Podunk, TX is increasing at the astounding rate of 38% per year. The population is
currently 6342. What will the population be in 5 years? 13. If a paper whose thickness is .004 in is folded until it is more than 2.3 inches thick, how many times was it
folded? 14. If a .0003mg microorganism’s size triples every week, how long until the organism has a mass of 14 mg?
25
PAP Algebra II Unit 7B Review Name: ___________________
1. Is it better to invest your money at 5.5% interest compounded continuously or at 5.8% interest compounded monthly if you have $12,000 to invest for 4 years?
2. If 16 32a b , what is a
b?
3. If you have an account that has an interest rate of 1.9% compounded monthly,
how long will it take for your money to triple?
4. If $1000 becomes $2578 with interest compounded continuously for 12 years, what was the interest rate?
5. The model for the Richter Scale is 0
logI
RI
where 0 1I .
a) Find the intensity of a 6.8 quake.
b) If the intensity of a quake is 5,640,000, what is the Richter scale value?
c) How many times more intense is a 6.3 quake from an 8.7 quake?
26
6. Mosquitoes are tripling in number each week. If there are currently 300 mosquitoes in your bug zapper in the back yard, when will there be 2000 mosquitoes?
Graph the following functions. Be sure to label at least 2 points and an asymptote.
7. 3xy 8. 3logy x
Domain: ________________ Domain: __________________ Range: ________________ Range: ___________________
9. 53 2xy 10. 32log ( 4)y x
Domain: __________________ Domain: __________________
Range: _________________ Range: __________________
27
11. If 3( ) log 2 1 f x x , find 1( )f x .
12. If the population of deer in the forest preserve is currently 1200 and is growing at a rate of 12% per year, what is the domain for the function if the population is monitored until it reaches 3000?
13. Expand the following completely:
a) 24
7log3
xy
z
b) 2
34
lnx
y
14. Condense into a single log:
a) 1 12 4
ln 3ln ln ln( )g f h r y
b) 13(2log 4log log 5log )p q r s
28