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1
Honors Math 3 Unit 4 Exponential and Logs NAME ___________________________
Day Date Topic(s) /Classwork Homework
1
Solving Radical Equations p. 2, odds p. 17 p. 3, evens
2 Rules for Domain Inverses of Functions
p. 8: #’s 1-8
3
Graphing Exponentials with transformations p. 11-12: #’s 1-8
4 Solving exponential equations with like bases More Practice with Laws of Exponents (Advanced)
p. 13-14: Do evens
5 Quiz 1 (Days 1-4) Introduction to Logs, Graphing logs
p. 16: #’s 1-6
6
Exponential Form to Log Form (vice versa) Evaluating logs Solving Log Equations (single logs)
p. 18: #’s 1-22
7 Group Quiz Test Review: p. 20-22
Finish p. 20-22 p. 34: Do problems with *
8 Unit 5 Test (Part 1) Delta Math HW Online
9 Properties of Logs Solving Log equations
p. 26 #’s 1-40
10 Natural Logs, Common Logs Solving with base e Odds on p. 29-30
Evens p. 29-30
11 Quiz 2 Change of Base Solving with Unlike bases
p. 31: #’s 1-25
12 Applications and review p. 33: #’s 1-8
13 Group Quiz Review: p. 34-37
p. 34-37
14 Unit 5 TEST (Part 2) Delta Math Online Problem Set
2
Day 1 CW: Solving Radical Equations: EXTRANEOUS SOLUTIONS: answers that do not work when put back into original problem. *only check for extraneous solutions when variable is under radical and index is even *If variable is under radical, isolate radical and raise both sides to the index value. *If variable in NOT under radical, isolate variable, simplify.
1. 𝑥 + 1 = 𝑥√5 2. √2𝑥 + 3 = 𝑥
3. √4𝑥 + 28 − 3√2𝑥 = 0 4. 2𝑥 + 7 = −𝑥√2
5. √7𝑥 − 3 − 2𝑥 + 3 = 0 6. √𝑥 + 8 − √𝑥 + 35 = −3
7. √2𝑦 − 3 − √2𝑦 + 3 = −1 8. √𝑦 − 53 − 2 = −4
9. √𝑥 + 11 − √15 + 2𝑥 = 1 10. −2√2𝑥 − 15
+ 4 = 0
3
Day 1 HW: Solving Equations Containing Radicals Solve each equation. (#4 – 16 Check answers for extraneous solutions! Except #5,10)
1. 0537 x 2. 634 xx
3. 2318 xx 4. 058 x
5. 473 y 6. 0234 x
7. 2158 n 8. 6841 t
9. 712274 v 10. 32563 u
11. 10246 xx 12. 20749 uu
13. 39 kk 14. 8610 xx
15. 972 xx 16. 052234 2 xxx
4
Day 2 Notes: RULES FOR DOMAIN: Write answers using interval notation. (1) If x is in the numerator and raised to a positive integral exponent the domain: (−∞, ∞) all reals a. f(x) = x2 D: _______________________________________
b. f(x) = 8
4
3
x
D: _______________________________________ (2) If x is in the denominator, x cannot be any value that will make the denominator zero.
a. f(x) = x
1
D: _________________________________________
b. f(x) = 3x
x
D: __________________________________________
c. f(x) = )4)(2(
1
xx
x
D: __________________________________________
(3) If x is inside an even root, values of x are restricted to ones that will make the radicand 0.
a. f(x) = x D: ____________________________________________
b. f(x) = 3x D: ____________________________________________
c. f(x) = √2𝑥 − 54
D: ____________________________________________
d. 𝑦 = √2 − 4𝑥 D: ____________________________________________ (4) If x is in an even root and in the denominator, values of x are restricted to the one that will make the radicand > 0.
a. f(x) = x
1
D: _____________________________________________
b. f(x) = 5x
x
D: ______________________________________________
c. 𝑦 =𝑥+3
√7−𝑥4 D: _____________________________________________
Find the Domain.
1. 𝑦 = √3𝑥3
2. 𝑦 = 1
𝑥2 + 1 3. 𝑦 =𝑥−2
𝑥2−2𝑥−15
5
Day 2 CW: Inverse Functions
Examples:
1) Given the t-chart for 𝑓(𝑥), find the t-chart for 𝑓−1(𝑥). 2) Given graph of 𝑓(𝑥), graph 𝑓−1(𝑥).
Find the inverse.
3) 2)( xxf 4) 𝑓(𝑥) = 𝑥2 + 3 5) 𝑦 = 3𝑥2 − 2
Graph the following function, find the domain and range, then graph its inverse and find the domain and range.
6) 𝑦 = √𝑥
D: ________________
R: ________________
𝑓−1(𝑥) = _____________
D: ________________
R: ________________
)(xf )(1 xf
x y x y
-5 0
-3 -2
1 5
2 -1
3 3
5 6
6
7) 𝑦 = √𝑥 + 3
D: ________________
R: ________________
𝑓−1(𝑥) = _____________
D: _______________
R: ________________
8) 𝑦 = √𝑥 + 3
D: ________________
R: ________________
f-1(x) = _____________
D: _______________
R: ________________
Find the inverse of 𝑓(𝑥).
9) 𝑓(𝑥) = 𝑥3 + 2 10) 𝑓(𝑥) = 𝑥2
3 11) 𝑓(𝑥) = (𝑥 − 4)2
7
Find the inverse of each function and graph both the function and its inverse on the same grid. Then state
the domain of both the function and its inverse.
12. 𝒇(𝒙) = √𝒙 − 𝟑 D:_____________ R:______________
𝒇−𝟏(𝒙) = _______________ D:_____________ R:______________
13. 𝒇(𝒙) = −𝒙𝟐 − 𝟐 D:_____________ R:______________
𝒇−𝟏(𝒙) = _______________ D:_____________ R:______________
8
Day 2 HW. Find the inverse of each function.
1. 𝑦 = 𝑥2 − 9 2. 𝑦 = (3 − 𝑥)2 − 1 3. 𝑦 = √𝑥 − 1 + 3
4. 𝑓(𝑥) = 3𝑥2 − 5 5. 𝑓(𝑥) = 2𝑥2
5+ 1 6. 𝑓(𝑥) = √𝑥 + 2 − 4
Find the inverse of the given function and graph both the function and its inverse on the same grid. Then
state the domain of both the function and its inverse.
7. 𝒇(𝒙) = √𝒙 − 𝟒
D:_____________ R:______________
𝒇−𝟏(𝒙) = _______________
D:_____________ R:______________
8. 𝒇(𝒙) = 𝒙𝟐 + 𝟑
D:_____________ R:______________
𝒇−𝟏(𝒙) = _______________
D:_____________ R:______________
9
Day 3 CW: Graphing Exponential Functions
Complete the tables, then graph.
Ex1. y = 2x
Ex 2. y = 4x
Ex 3. y = (1/2)x
𝑦 = 𝑎(2)𝑏(𝑥−ℎ) + 𝑘
If a is negative: Reflect over the __________ If b is negative: Reflect over the _____________
h: __________________ or _______________ k: ___________ or _____________
|a| > 1 Vertical ______________ 0 < |a| < 1 Vertical _______________
x y= 2x Y
-4
-3
-2 y = 2-2 ¼
-1 y = 2-1 ½
0 y = 20 1
1 y = 21 2
2 y = 22 4
3
4
5
Domain: Range: Y-intercept: X-intercept: Asymptote:
x y= (1/2)x y
-2 -1
0 y = (1/2)0 1
2
Domain: Range: Y-intercept: X-intercept: Asymptote:
x y= (4)x y -3
-2
-1 0 y = (4)0
1 2
3
Domain: Range: Y-intercept: X-intercept: Asymptote:
10
Day 3 Examples: Graph, describe transformations, state the domain & range
1. 12 xy Transformation: 2. 12 xy Transformation:
3. 32 2 xy Transformations: 4. xy 2 Transformation:
5. xy
2 Transformations: 6. x
y 24 Transformation:
Domain: Range:
Domain: Range:
Domain: Range:
Domain: Range:
Domain: Range:
Domain: Range:
11
7. 221
xy Transformations: 8. x
y
23 Transformations:
9. xy 5 10. 25 1 xy Transformations:
Day 3 HW: Graphing Exponential Functions Worksheet
1. 13 xy 2. 23 xy
Domain: Range:
Domain: Range:
Domain: Range:
Domain: Range:
Domain: Range:
Domain: Range:
12
3. xy
3 4.
y 2x2 3
5. 42 x
y 6. xy
32
7. 423
xy 8. 132
2
xy
Domain: Range:
Domain: Range:
Domain: Range:
Domain: Range:
Domain: Range:
Domain: Range:
13
Day 4 CW/HW: I. Solve. 1. 53 = 52x – 1 2. 9x = 3x + 4 3. 5x – 1 = 1252x + 3 4. 23x – 1 = 4x + 2
5. 82x – 2 = 42 – x 6. 1
3
33
1
x
x
7. 3
42
49
17
x
x 8. 63
3
25
1125
x
x
9. 33
8
1
32
1
xx
10. 4
12
2
18
x
x 11. 4x = 2x + 3 12. 53x – 1 = 25 x + 4
13. 3x – 1 = 27 2x +3 14. 1252x – 2 = 253 – x 15. 1
3
66
1
x
x
16. 3
42
100
110
x
x
17. 63
3
4
18
x
x 18. 42
8
1
32
1
xx
19. 5
13
2
14
x
x 20. 252x = 5x + 6
21. 6x + 1 = 36x – 1 22. 10x – 1 = 1004 – x 23. 27
13 x 24. 1255 x
25. 82 + x = 2 26. 4 1 - x = 8 27. 272x – 1 = 3 28. 7749 2 x
14
II. Simplify & express with positive exponents.
1. 2
3 2. 04 3. 822
4. 2225 5. 32
323
6. 21215
7. 3232 33 8. 1515 55 9. 1515 22
10. 22121 33 11. 2
82ba 12. 333 yx
13. (𝑥√3 − 𝑦√2)2 14. (𝑥−√8𝑦√50)√2
15. −32𝑥
45𝑦
16
4𝑥15𝑦
56
16. (𝑥3√2𝑦−2√3)
−3
𝑥4√2𝑦−4√3
15
Day 5 CW: Logarithmic Function: inverse of an exponential function Find the inverse of 𝑦 = 𝑏𝑥 Example 1: Graph on the same axis. Make a table for each.
a) y = 2x
b) xy2
log
1. )(log 22
xy 2. 22 xy log
Domain: Range: Asymptote:
Domain: Range: Asymptote:
16
Day 5 HW: Graphing Log Functions Worksheet
1. xy2
log 2.
y log2(x 3)
3.
y log2 x 3 4.
y log2(x 1) 2
5.
y log4 x 1 6.
y log4 (x 2) 3
Domain: Range: Asymptote:
Domain: Range: Asymptote:
Domain: Range: Asymptote:
Domain: Range: Asymptote:
Domain: Range: Asymptote:
Domain: Range: Asymptote:
17
xyb log ybx
Day 6 CW: Find the inverse of 𝑦 = 2𝑥.
y is called the logarithm, base 2 of x. It is written in “log form” as: A LOGARITHM is AN _______________________. Exponential Form: 𝐵𝑎𝑠𝑒𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 = 𝐴𝑛𝑠𝑤𝑒𝑟 Logarithmic Form: 𝑙𝑜𝑔𝑏𝑎𝑠𝑒(𝑎𝑛𝑠𝑤𝑒𝑟) = 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡
* Base will ALWAYS be ______________ (never = to zero, never = 1, never negative)
** Answer will ALWAYS be _______________ (never = to zero, never negative)
*** Exponent can be _________ real number
Find the exact value without using a calculator. (What power of ___ gives you ____?)
1) log5 25 2) log2 32 3) log3 27 4) log5 251
5) log5 1 6) log5 5m 7) log2 64 8) log8 64
9) log23 2 10) log2 8
1 11) log2 2 12) 1log2
A few rules about the values of b and x:
13) )2(log2 14) 2log 2 15) 2log0 16) 0log2
17) 4
2 2log 18) 𝑙𝑜𝑔5(5−3) 19) lit77log 20) 3
99
mlog 21) 7
22log
Solving logarithmic equations:
22) 213log5 x 23) 38log x 24) x9log27
25) 1316log2 x 26) 481log x
Note: log b x
Exponential Form: 32 = 9 Logarithmic Form: log 3 9 = 2
18
Day 6 CW Exercises I. Write in exponential form.
1. 38log 2 2. 481log3 3. 13
1log 3 4. 2
25
1log 5
5. 2
12log 4 6.
4
38log16 7. 01log8 8.
3
24log8
9. 481log
3
1 10. 2100log10
II. Write in logarithmic form.
1. 2525 2. 4381 3. 160 4. 283
1
5. 9
13 2 6.
64
116 3
2
7. 12525 2
3
8. 93
12
9. 32
12 5
III. Find the logarithm. 1. 1.0log10 2. 01.log10 3. 3log 27 4. 9log 27
5. 16log8 6. 2
1log 4 7.
4
1log
2
1 8. 9log3
1
9. 25log5
1 10. 10
2 2log
IV. Find x. 1. 481log x 2. 110log x 3. 216log x 4. 3125log x
5. 4log 4 x 6. 2log6 x 7. 0log3 x 8. 3
2log8 x
9. x100log10 10. x27log9 11. x8log16 12. x5.log 2
13. 4log2
x 14. 6log3
x 15. 2
13log x 16.
6
12log x
19
Day 6 HW: Write each equation in logarithmic form.
1. 53 = 125 2. 8127 3
4
Write each equation in exponential form.
3. log100.00001 = -5 4. 2
1
3
6log
2
3
Evaluate each expression. 5. 81log3 6. 0001.0log10
7. 16
1log 2 8. 27log
3
1
9. 1log 9 10. 4log8
Solve each equation.
11. 2
3log 4 x 12. 416log y
13. 38
1log a 14.
2
1log 7 n
15. 3
4log
5y 16.
6
19log 3 x
17. 47log73log 88 xx 18. 6log208log 77 xx
19. 164log19log 33 xx 20. 133log9log 1212 xx
21. 6log30log 5
2
5 x 22. xx 5log6log 4
2
4
20
Day 7 Unit 3 Part 1 Test Review:
Evaluate the following:
1. 625log5 2. 64log2 3. 1log4 4.
000,100log10 5. 2.0log5
Simplify. (No negative or zero exponents)
6. 𝑦3
2𝑥2 ∙8𝑥7
𝑦9 7. (2𝑦3
𝑥2 )5
8. 6𝑥−5𝑦−3
8𝑦−7
Find the inverse of the following:
9. 𝑓(𝑥) = 9𝑥2 − 13 10. 𝑦 = √𝑥 − 5 11. 𝑓(𝑥) =𝑥2−3
7
Express in logarithmic form.
12. 25𝑘 = 100 13. (1
3)
−3
= 27
State the transformations of each of the following from the parent graph 𝑓(𝑥) = log2 𝑥.
14. 𝑔(𝑥) = log2(𝑥 + 4) − 9 15. 𝑔(𝑥) = − log2(𝑥 + 5)
16. 𝑔(𝑥) = log2(𝑥 − 3) + 6 17. 𝑔(𝑥) = log2(−𝑥) + 6
Solve for x.
18. 43x + 1 = 8x – 1 19.
63
3
4
18
x
x
21
Solve each equation. Show all work and circle final answers. No decimal answers.
20. log125 𝑥 =2
3 21. log𝑥 3 =
1
4
Find the inverse of the given function and graph both the function and its inverse on the same grid. Then
state the domain of both the function and its inverse.
22. 𝑓(𝑥) = √𝑥 − 4
D:_____________ R:______________
𝑓−1(𝑥) = _______________
D:_____________ R:______________
23. 𝑓(𝑥) = 𝑥2 − 5
D:_____________ R:______________
𝑓−1(𝑥) = _______________
D:_____________ R:______________
22
Graph and analyze the following:
24. 𝑦 = 2𝑥+4 − 1
Transformation: ______________________________
Domain : ______________ Range: _____________
25. 𝑓(𝑥) = log2(𝑥 − 3) + 5
Transformation: _______________________________
Domain : _____________ Range: ________________
23
Day 9 CW: Properties of Logarithms:
Condensed Form Expanded Form
Product Property mnb
log
nm
bbloglog
Quotient Property n
mb
log
nmbb
loglog
Power Property
logb mp
mp
blog
Equality Property: nmthennmIf bb loglog
Examples: Use the properties of log to expand each log:
1.
log3xy 2.
log5x
y
3.
log7a2b 4.
log9a2
cb3
Use the properties of log to write each as a single log:
________________________5.
log2 x log2 y log2 z
________________________6.
3log2 x log2 y 4log2 z
________________________7.
2log xy log5 3(logz log2)
Proof of the Product Proof of the Quotient Property: Property:
nmmn
yxmn
mnb
mnbb
nbandmbthen
nyandmxLet
bbb
b
yx
yx
yx
bb
logloglog
log
loglog
nmn
m
yxn
m
n
mb
n
m
b
b
nbandmbthen
nyandmxLet
bbb
b
yx
y
x
yx
bb
logloglog
log
loglog
24
Evaluate: (no calculator)
20. 733log
21.
log3 3x 22.
9log3 5 23.
log3 94
Given log2 5 = m. Write the answer in terms of m.
24. Find log2 20 25. Find log225 = Given log12 9 = a & log12 18 = b. Write the answer in terms of a and b.
26. Find log12 (4
3) 27. Find log12216
Given log12 8 = k & log12 10 = p. Write the answer in terms of k and p.
28. Find log1280 29. Find log12 960
Since logarithmic functions and exponential functions are
inverses xax
a log and xaxa
log
You Always Know…
Loga ax = x Ex. Log3 35 = 5 because in exp. form
35 = 35
Loga a = 1 Ex. Log4 4 = 1 because in exp. form
41 = 4
Loga 1 = 0 Ex. Log7 1 = 0 because in exp. form
70 = 1
25
Day 9 CW: I. Express in expanded form using the properties of logs.
1. 𝑙𝑜𝑔37𝑥2 2. 𝑙𝑜𝑔48
√𝑥
3. 𝑙𝑜𝑔6√𝑥5
𝑦3 4. 𝑙𝑜𝑔5(2𝑎 ∙ 3𝑏)
II. Express as a single logarithm with coefficient of one. 5. 𝑙𝑜𝑔3𝑥 − 2𝑙𝑜𝑔3𝑦 6. 3𝑙𝑜𝑔3𝑎 + 4𝑙𝑜𝑔3𝑏 − 𝑙𝑜𝑔3𝑐 7. 𝑙𝑜𝑔2(𝑥 + 2) − 𝑙𝑜𝑔2(𝑥 − 1) 8. 𝑙𝑜𝑔(2𝑥 + 1) + 𝑙𝑜𝑔(3𝑥 − 2) 9. Given that 𝑙𝑜𝑔34 = 𝑥, 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑙𝑜𝑔316 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑥.
10. If 𝑙𝑜𝑔𝑚𝑎 = 3 𝑎𝑛𝑑 𝑙𝑜𝑔𝑚𝑏 = 4, 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑙𝑜𝑔𝑚 (1
𝑎𝑏)
11. If log x = 3 evaluate log (10x2) 12. If log 6 2 = x and log 6 5 = y, express each logarithm in terms of x and y. a. log 6 40 b. log 6 3 Solve for x. 13. 12)(x
4log1)(2x
4log 14. 𝑙𝑜𝑔72𝑥 − 𝑙𝑜𝑔75 = 𝑙𝑜𝑔74
15. 2𝑙𝑜𝑔7𝑥 = 4𝑙𝑜𝑔72 + 2𝑙𝑜𝑔75 16. 𝑙𝑜𝑔3(𝑥 + 8) = 2 − 𝑙𝑜𝑔3𝑥
17. 𝑙𝑜𝑔2(𝑥2 − 2) − 𝑙𝑜𝑔2 (1
2𝑥 + 5) = 1 18. 𝑙𝑜𝑔25 + 2𝑙𝑜𝑔23 = 𝑙𝑜𝑔2𝑥 + 𝑙𝑜𝑔2(𝑥 − 4)
Laws of Exponents:
. number real any for,loglog
iffloglog
logloglog
logloglog
kMkM
NMNM
NMN
M
NMMN
bk
b
bb
bbb
bbb
26
Day 9 HW: NO CALCULATOR! Evaluate:
1. 813
log 2. 00010.log 3. 16
12
log 4. 525
log 5. 273
1log
6. 19
log 7. 48
log 8. 37log57log7
9. ))9(log(loglog 3210
Expand using the properties of Logs:
10. 27 3log x 11.
36logbc
a
Write as a single logarithm with a coefficient of 1.
12. cba 222 log5log3log 13. )log2log3(7
155 yx
Evaluate:
14. 53log23 15. 7log aa 16.
35log225log35 17.
122log62log22
18. 53log
9 19. 32log
8
Given: a3
2log and b5
2log find the following in terms of a and b:
20. 1802
log 21. 4502
.log 22. 25
62
log
Solve.
23. 216 x
log 24. 3
28
xlog 25. x255
log
26. 2
13
xlog 27. x8
16log 28. 2372
555logloglog x
29. 221 )log(log xx 30. 112 )log()log( xx 31. 416 xlog
32. 6
193
xlog 33. xx 56
4
2
4log)(log 34. 2
6xlog
35. x52.log 36.
4
13
xlog 37. 24
3 )(log x
38. )loglog(log 22426
1x 39. x
8
12
log 40. )(log)(log 51177
xx
27
Day 10 CW: Natural Log
Consider the following: y =
n
n
11
What happens as n approaches infinite? Let n = 10 y = n = 100 y = n = 1000 y = n = 10,000 y = n = 100,000 y = Round to three places: y = “e” _____________ “e” is the base for natural logs
Instead of using elog the symbol “ln” is used for log with base e.
Rewrite in Log or exp. form:
1. ln x = 5 2. 38972 .e 3. 12xe 4. ln 1 = 0 Simplify:
1. e-3 . e5 2. (3e4x)2 3.
32
4
1
e 4. xe24
5. ex . 4e2x+1 6. 3 964 xe
Solve: (round to 4 places) 1. e-x = 6 2. 1.2e-5x + 2.6 = 3 3. 0.25x - 0.5 = 2
4. 43
143
2xe
28
Day 10 CW/HW: I. Express in logarithmic form:
1. 102.33 = x 2. 10 x = 379.31 3. e2.5 = 12.18
4. 5
1
e = 1.221 5. e-2 = 0.135 6. 649.1e
II. Express in exponential form: 7. Log 229 = 2.3598 8. Log .8033 = -.0951 9. ln 8 = 2.079
10. 39.14
1ln 11. ln .01 = -4.605 12. ln e = 1
III. Express as a single Logarithm.
13. ln 6 + ln 5 – ln2 14. 6ln9ln2
1 15. 38ln
3
1 16. 24ln
2
3
IV. Find the Log of each number using a calculator. (Round to four decimal places) 17. Log 95 18. Log .233 19. Log 6.437 x 10-9 20. Log 1.8519 21. ln 120 22. ln 690
23. 5
6ln 24. 3
2
84ln 25. ln 0
V. Use a calculator to find the antilog ‘x’ of each Log or Natural Log. 26. Log x = 2.63 27. Log x = -.4089 28. Log x = 5.3 29. ln x = 2.208 30. ln x = 1.808 31. ln x = -.105
29
VI. Evaluate WITHOUT USING A CALCULATOR! 32. ln e 33. ln ( e2 ) 34. ln 1
35. ln 0 36. ln e
1 37. eln
38. ln (en ) 39. 7ln6ln e 40.
7ln2 e 41. 9ln
2
18ln
3
1
e
VII. Simplify and then Evaluate. (Use a calculator only when necessary)
42. ln 48 – 4 ln 2 43. 3ln212ln9ln2
1 44. 3ln25ln45ln
2
1
45. 6ln8ln e 46. ln 6 + ln 30 – (ln 5 + 3 ln 2) 47. )3ln2ln5(8ln4ln
2
1
48. 3 ln 4 – (ln 2 + ln 8 ) 49. )3(ln
2
1
e
VIII. Solve and Check! 50. Log (x + 3) + Log (x) = 1 51. Log (x + 3) - Log ( x – 1) = 1 52. Log x = 2 - Log (x + 21) IX. Solve by rewriting the problem in another form. Round decimal answers to four places!
53. ln x =3 54. 3
1ln x 55. 5xe
56. 5
3
ln ex 57. 12ln xe 58. 2
1ln x
30
Day 11 CW:
Change of base formula:
logb m logn m
logn b
Round all answers to 4 places.
To solve, PUT IN OTHER FORM. Or Take the log of both sides
1. log4 22 2. log12 95
3. 4x = 24 4. 7x = 20
5.
13x 5x2 6.
7x2 53x
Use logarithms to solve each equation. Round to 4 places. Do not use logs for #11,12 1. 3.5 x = 47.9 2. 8.2 y = 64.5 3. 7.2 a – 4 = 8.21 4. 2 b + 1 = 7.31
5. 5.78log3y 6. 8.91log 4k
7. 4 2x = 9 x – 1 8. 7 3b = 12 b + 2 9. 6 x – 2 = 4 3 – x 10. 3 4r = 5 r – 1
11. 4417 3
2
c 12. 1117 8
9
x
13. 30log7x 14. 152log3n
15. 725 32
x 16. xx 734 54
31
Day 11 HW: PRACTICE EXERCISES: Solve:
19) log (x+9) – log x = 1 20) log4(x+3) + log4(x-3) = 2 21) log5(x+4) +log5(x-4) = 2
22) xx loglog3
23) xx loglog4
24) 11log 25 x
25) 33 43 2 2logloglog xx
Key: 1) 3 6) –1 10) 1.1073 14) 3.5807 18) 1/3 22) 1, 109 2) 5 7) –3, -1 11) 2.7095 15) 5.6456 19) 1 23) 1, 1016
3) 3.3219 8) –3, 0.5 12) 7.4520 16) 10 20) 5 24) 2 6
4) 5.0444 9) 1.4037 13) 3.6067 17) 1 21) 41 25) 42
5) 5/2
1) 2x = 8 2) 2x = 32 3) 2x = 10 4) 2x = 33 5) 54x - 7 = 125 6) 43x + 5 = 16
7) 27
13 42
xx
8) 2793
25 xx
9) 4x = 7 10) 8x = 10 11) 2x = 3x - 1 12) 3x + 2 = 5x - 1 13) (2.8)x = 41 14) (3.4)x = 80 15) (1.7)x = 20 16) log x + log (x–9) = 1 17) log x + log (x+9) = 1 18) log x – log (x+3) = -1
32
Day 12 CW: FORMULAS:
Compound Interest :
tn
n
rPA
1 Continuously Compounded :
trPeA
“n times a year “
Value of an Asset : trPV 1 Growth &/ Decay : tkney
CLASS EXAMPLES: 1.) How long would it take for an investment of $2500 to triple if it is invested in an account that earns 6% interest compounded quarterly.
2.) Your bank promises to double your money in 2
18 years. Assuming the interest rate is compounded
continuously, what is the interest rate? 3.) The half-life of a radioactive isotope is 9 years. Find the constant “k” for a 20 gram sample. 4.) Zeller industries bought some equipment for $50,000. It is expected to depreciate at a steady rate of 10% a year. When will the value be half the original value? 5.) The Jamesons bought s new house for $144,500 five years ago. The home is now worth $187,850. Assuming a steady rate of growth , what was the yearly rate of appreciation?
33
Day 12 HW: Applications of Logarithms Solve each problem. Show all work! 1. 2. 3. 4. 5. 6. 7. 8.
Suppose $500 is invested at 6% annual interest compounded twice a year. When will the investment be worth $1000?
Suppose $500 is invested at 6% annual interest compounded continuously. When will the investment be worth $1000?
An organism of a certain type can grow from 30 to 195 organisms in 5 hours. Find k for the growth formula.
For a certain strain of bacteria, k is 0.825 when t is measured in days. How long will it take 20 bacteria to increase to 2000?
An investment service promises to triple your money in 12 years. Assuming continuous compounding of interest, what rate of interest is needed?
A substance decomposes radioactively. Its half-life is 32 years. Find the constant k in the decay formula.
A piece of machinery valued at $250,000 depreciates at 12% per year by the fixed rate method. After how many years will the value have depreciated to $100,000?
Mike bought a new car 8 years ago for $5400. To buy a new car comparably equipped now would cost $12,500. Assuming a steady rate of increase, what was the yearly rate of inflation in car prices over the 8-year period?
34
Review Sheet. NO CALC, unless rounding is stated *1. Write an equivalent logarithmic equation for x5 = 32. 1. _______________________
*2. Write an equivalent exponential equation for 5ln x 2. ______________________
*3. What is the domain and range of f(x) = ex + 2? 3. D: _____________ R: ____________ 4. The log 0.034 is between what two consecutive integers? 4. ______________________ 5. Given log 8.1 = a, find the log 8100 in terms of a. 5. ______________________ *6. Given f(x) = 3x and g(x) = x9log , find f-1( g (9)). 6. ______________________
7. Write as a single logarithm. 2
ln1 x 7. ______________________
Simplify Completely:
*8. 64
1log8 9. ln e9 *10. 2
3
2 4log *11. 4log8 12. 5
3log
27
Solve the following: *13. 01log x *14. x100log1000 *15. x354 log)25(loglog
*16. 181824 x *17. 0))(log(loglog 32 x
18. 2
ln54 xex *19. 68log x
*20. x7
8
1 16log *21. 3
2
25
1log x
*22. 7log)3(loglog 92 x 23. 2)2(loglog2 33 xx
35
24. )5log2(log220loglog2
17777 x
25. )6log3log2(log2
3)5log215(log2log 33 x
26. If a5log find log 0.04 in terms of a.
27. If 2log3c and 5log3d , express 5
104log 3 in terms of c and d.
Solve. Round to 4 places.
28. 131 5.46 xx 29. 5.74)1(2 3
4
x
30. 0103732 xx 31. 3.35.21
44
xe
36
Decay/Growth Value of an Asset Compounded continuously Compounded “n” times/year
y = nekt V = P ( 1 r)t A = Pert tn
n
rPA
1
32. To the nearest dollar, what amount must be invested at 6% compounded continuously for 14 years in order for a balance to be $23,140? 33. A tractor that 4 years ago cost $8,000, now is worth only $3200. Find the average annual rate of depreciation. 34. The population of a certain colony of bacteria doubles every 5 hours. How long will it take for the population to triple? 35. Radioactive iodine is used to determine the health of the thyroid gland. It decays according the formula y = ne -.0856t where t is in days. Find the *half-life of this substance. (*time it takes for a substance to be half its original mass)
36. A radioisotope is used as a power source for a satellite. The power output is given by 25050
t
eP
where P is the power in watts and t is time in days. a) Find the power available after 100 days. b) If ten watts of power are required to operate, how long can the satellite continue to operate? Graph on graph paper: 37. y = 2x-2 -1 38. yx
2
1log
Solve for x.
39. xx 55 40. 15 xx 41. 2310 xx
37
Extra Review Use ONLY a (4 function) calculator . Write in logarithmic form.
1. 72 = 49 2. x6
1
5 Write in exponential form.
3. 2819
log 4. 532
12
log
Solve for x.
5. 34
xlog 6. 481xlog 7. 3
227
xlog 8. 28
xlog
Write in expanded form.
9. 6
2
5
xlog 10.
3
45xylog 11.
5 4axlog
Write as a single logarithm with a coefficient of 1.
12. 632777
logloglog xy 13. 5123bbb
xy log)(loglog
Solve for x.
14. 732999
logloglog x 15. 392666
logloglog x
16. 2644
)(loglog xx 17. 382
2 )(log x
If log12 8 = a and log12 10 = b, find the following in terms of a and/or b:
18. 80012
log 19. 812
log
Solve for x: (give the calculator ready answer)
20.
5.3x 72