Ballenas Secondary Exponents and Logarithms Problems Homework

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Ballenas Secondary

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Hi there! This assignment is about exponents and logarithms. Please show all your work. I dont mind in which format the assignment is completed, you can print it and work on it or do it through a program online. Msg if you have any questions, thanks!

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WCLN PCMath 12 - Rev. 2019 Instructions: Name: _______________________________ UNIT 4 - LEARNING GUIDE: Logarithms Using a pencil, complete the following questions as you work through the related lessons. Show ALL work as it is explained in the lessons. Do your best and ask your instructor if you do not understand any questions! Lesson 1 – Review the Exponent Laws 1. Simplify the following and write answers with positive exponents only. a) �x2 � 4 b) x−3 x2 2 2y3 d) � y � � g) (xy)2 � x 1 2x3 � −1 � −2 ÷y −1 �m2 n3 � m3 n 9y6 e) �4x2 � 1� 2 (2x)−2 c) (3x2 )3 ÷ xy f) h) � Page 1 of 19 �2m2 � n3 (3m)−1 (2n)2 −1 2 n �m 2 � m 3 4 � ÷ �3n3 � WCLN PCMath 12 - Rev. 2019 2. Simplify and then evaluate the expressions for x = –1, y = 2, z = 3. a) (3x 2 y −1 z 2 )(4x −1 y 2 z) b) �2xy3 z−1 � x 2 y4 2 Lesson 2 – Solving Equations Involving Exponents 1. Solve for x using the reciprocal of the exponent. 2 a) x 3 = 4 3 c) 5x 2 = 40 1 b) x −2 = 9 2 d) (4x + 1)3 = 25 Page 2 of 19 WCLN PCMath 12 - Rev. 2019 2. Solve for x by converting to common bases. Show all your work. 1 x a) 2x = 32 b) �3 � = 81 c) 5x = 625 d) 3x−2 = 27 e) 42x = 8 f) 27x−1 = 9 g) 3(16x+2 ) = 96 h) 252x−1 = 5x i) 2(8x+1 ) = 16 j) 4x = 8x+1 k) 33x+4 = 9x l) �4 � Page 3 of 19 1 −x+2 = 83x−1 WCLN PCMath 12 - Rev. 2019 Lesson 3 – Defining a Logarithm 1. Convert the following exponential equations to logarithmic form. a) 40 = 1 b) 52 = 25 1 d) ab = 𝑐𝑐 c) 92 = 3 2. Convert the following logarithmic equations to exponential form. a) log 3 9 = 2 1 b) log 4 16 = −2 c) log 0.1 = −1 d) log a b = c 3. Evaluate each of the following. a) log 1000 b) log 2 16 1 c) log 9 3 d) log 2 8√2 4. Solve for x by converting to exponential equations and then simplifying. a) log 4 64 = x b) log x 9 = 2 c) log 5 x = 3 d) log x 27 = 2 3 Page 4 of 19 WCLN PCMath 12 - Rev. 2019 5. Briefly explain the restrictions on the domains of logarithmic functions and then state the domain for each of the equations below. a) y = log 3 (x + 5) b) y = log 2 (2x − 3) c) y = log x (3 − x) Lesson 4 – Laws of Logarithms 1. Write each of the following as a single logarithm. a) log x + log y b) 2log 3 x − log 3 y 1 c) log x + 5 log y − 3 log x 1 d) − log x + 2 log y + 1 e) 4 log 2 x + 3 log 2 y − 3 2. Use the logarithm laws to expand each of the following. x a) log �y2 � b) log 2 (√xy) Page 5 of 19 y 2 c) log �x� WCLN PCMath 12 - Rev. 2019 100x d) log �z2 5 � e) log 2 � �y 3 �x2 y 16 � 3. Given that log a x = 2 and that log a y = 3, use the logarithm laws to expand the following and evaluate the exact value. ay a) log a (x 2 y) b) log a � x � c) 3log a �√xy� 4. Simplify each of the following. a) 52 log5 x b) 1 log3 2 1 + log 52 c) 8log2 x 5. Given that log 2 = a and that log 3 = b, determine an expression for the following in terms of a and b without using a calculator. a) log 6 30 b) log �16� c) log 1200 Page 6 of 19 WCLN PCMath 12 - Rev. 2019 6. If log 3 x = 10 then evaluate each of the following without a calculator. a) log 3 (9x) x2 27 b) log 3 � x � c) log 3 � 3 � 7. Given that log 9 4 = a and log 81 5 = b, find an expression for log 3 20 in terms of a and b without using a calculator. Lesson 5 & 6 – Solving Exponential and Logarithmic Equations & Introduction to Exponential Functions 1. Solve the following exponential equations for the exact value of x then evaluate your solution rounding to 4 decimal places. a) 3x = 11 b) 5x−1 = 7 c) 73x + 3 = 21 d) 1 = 2(4x ) − 5 Page 7 of 19 WCLN PCMath 12 - Rev. 2019 e) 3x+1 = 5x f) 5(23x−2 ) = 9x f) 62x+1 = 71−x 2. Solve the following algebraically for x. a) log 2 x = 0 c) log 15 − log 5 = log x b) log(x − 2) = 1 d) log 3 (x − 2) − log 3 2 = 3 Page 8 of 19 WCLN PCMath 12 - Rev. 2019 e) log x + log x = log 36 f) 2log 3 (1 − x) = 6 g) log(x − 1) + log(x + 2) = 1 h) log(2x + 1) = 1 + log(1 − x) i) log 2 (x 2 − 8) = 3 j) log 3 (x + 5) = 1 − log 3 (x + 3) k) log 2 (x − 3) − log 2 5 + log 2 (x − 2) = 2 l) log 2 (log 5 25) = x Page 9 of 19 WCLN PCMath 12 - Rev. 2019 Lesson 6, 7 & 8 – Applications of Exponential Functions Use the compound interest formula shown below to answer the first three questions. r nt A = P �1 + n� where r = rate in decimal form n = compounding per year (ex. daily is 365) t = time in years P = principal A = final amount You will need to be careful to keep at least 8 decimal places as you do your work in order to get an accurate answer. 1. Sarah invests $1200 at a rate of 5% per year compounded monthly. a) How much money would Sarah have after 7 years? b) How long does it take for the investment to grow to $2000? Give your answer to two decimal places. 2. An investment of $3000 lost 4% per year compounded weekly. a) How much is the investment worth after 5 years? Page 10 of 19 WCLN PCMath 12 - Rev. 2019 b) How long does it take for the investment to be reduced to half of the initial amount? 3. An initial investment of $900 that is compounded quarterly grows into $1400 in 12 years. What was the interest rate to one decimal place? Use the population formula shown below to answer questions 4 through 8. t A = A0 (1 + r)T where r = rate in decimal form t = time or distance A0 = initial amount A = final amount T = period of growth or loss Page 11 of 19 WCLN PCMath 12 - Rev. 2019 4. A population of 1200 badgers grows by 6% every 8 years. How long does it take the population of badgers to grow to 2000? Round to the nearest year. 5. The intensity of light decreases by 4% for each meter that it descends below the surface of the water. At what depth is the intensity of light only 20% of that at the surface? Round to one decimal place. 6. A colony of bees doubles in population every 5 weeks. How long does it take for the population to triple? Round to one decimal place. Page 12 of 19 WCLN PCMath 12 - Rev. 2019 7. The half-life of a certain radioactive isotope is 20 days. How long does it take for 50g of the isotope to decay to 10g? 8. A radioactive substance is produced from nuclear fallout. If 600 g of the substance decays to 250g in 20 years what is its half-life? Round to the one decimal place. 9. How many times more powerful is an earthquake with a Richter scale reading of 7.9 than an earthquake with a rating of 3.2? Page 13 of 19 WCLN PCMath 12 - Rev. 2019 10. An earthquake of 6.7 occurred off the coast of British Columbia. A few hours later, an aftershock occurred that was 150 times less intense as the original earthquake. What was the magnitude of the aftershock? Give your answer to one decimal place. 11. If celery has a pH of 5.9 then how many times more acidic are apricots if they have a pH of 4.1? Round your answer to one decimal place. Note that acidity = 10 −pH 12. Egg yolks have a pH of 6.4. What is the pH of egg whites if they are 63 times more alkaline? Page 14 of 19 WCLN PCMath 12 - Rev. 2019 Lesson 9 & 10 – Graphs of Exponential and Logarithmic Functions 1. The general equation for exponential functions is: y = a(bx+h ) + k. Which transformations will change each of the following and which letter would you alter to create these transformations. a) equation of the asymptote ________________________________________________ b) range ________________________________________________ 2. The general equation for logarithmic functions is: y = alog b (c(x + h)) + k. Which transformations will change each of the following and which letter would you alter to create these transformations a) equation of the asymptote ________________________________________________ b) domain ________________________________________________ 3. Determine the equation of the asymptote for each of the following functions. a) y = 5x+1 − 2 b) y = log(x + 4) − 3 c) y = −32x + 4 4. Graph the following equations. Explicitly state the equation of the asymptote, the x-intercept, the y-intercept, and the domain and range. a) y = 2x−3 − 4 Page 15 of 19 WCLN PCMath 12 - Rev. 2019 b) y = log 3 (x + 2) + 1 1 −x c) y = −3 �3� +6 Page 16 of 19 WCLN PCMath 12 - Rev. 2019 1 d) y = − 2 log 2 (x + 4) − 2 Page 17 of 19 WCLN PCMath 12 - Rev. 2019 SOLUTIONS Lesson 1 – Review The Exponent Laws 1. a) x11 f) 2. b) n a) –648 b) 9 a) 8 b) 2. a) 5 2. 3. 4. 5. h) 16 1. 1. c) g) 4x 8 y 12m7 g) − 1 m5 n4 3 d) 108x8 81n6 4m8 x5 e) 2y5 3y2 2x2 Lesson 2 – Solving Equations Involving Exponents 1 c) 4 d) 31 b) – 4 c) 4 d) 5 e) 2 i) 0 j) –3 k) – 4 h) 4 1 81 3 3 f) 4 b) log 5 25 = 2 2 a) 3 = 9 a) 3 a) 3 Answers may vary. b) 4 −2 = 1 c) log 9 3 = 1 c) 10 16 −1 1 2. 3. 4. 5. 6. 7. a) log(xy) x2 b) log 3 � � 7 2 y a) log x − 2 log y 1 y5 c) log � 2 � x 1 b) log 2 x +log 2 y 2 d) 2 + log x − 2 log z − log y 5 a) 7 b) 2 b) log 2 15 a) x 2 a) a + b b) b + 1 − 4a a) 12 b) –7 2a + 4b 10�y d) log � 2 x � 7 d) ac = b = 0.1 Lesson 4 – Laws of Logarithms 1. 1 d) log a c = b 2 c) − 2 c) 125 3 b) D: x > b) 4 b) 3 a) D: x > −5 3 l) − Lesson 3 – Defining a Logarithm a) log 4 1 = 0 5 e) log 2 � d) 2 d) 9 c) D: 0 < x < 3, x ≠ 1 4 √xy3 � 8 c) 2 log y − 2 log x 1 e) log 2 x + log 2 y − 4 3 c) 12 c) x 3 c) 2a + b + 2 c) 19 3 Lesson 5 & 6 – Solving Exponential and Logarithmic Equations & Introduction to Exponential Functions 1. Exact answers may vary d) 2. log 3 log 4 a) 1 ≅ 0.7925 b) 12 a) e) c) 3 d) 56 log 11 log 3 ≅ 2.1827 log 3 log 5−log 3 e) 6 ≅ 2.1507 f) –26 g) 3 b) f) h) log 5+log 7 log 5 ≅ 2.2091 2 log 2−log 5 3 log 2−log 9 3 4 ≅ 1.8945 i) ±4 j) –2 c) g) log 18 3log 7 log 7−log 6 2 log 6+log 7 k) 7 Lesson 6, 7 & 8 – Applications of Exponential Functions 1. a) $1701.69 2. a) $2456.00 3. 3.7% 8. 15.8 years b) 10.24 years b) 17.32 years 4. 70 years 9. 50118 5. 39.4 m 10. 4.5 6. 7.9 wks 11. 63.1 Page 18 of 19 ≅ 0.4951 7. 46.4 days 12. 8.2 l) 1 ≅ 0.0279 WCLN PCMath 12 - Rev. 2019 Lesson 9 & 10 – Graphs of Exponential and Logarithmic Functions 1. 2. 3. 4. a) vertical translation (k) a) horizontal translation (h) a) y = −2 b) x = −4 a) asymp: y = −4 x=5 31 y = − ≅ −3.9 8 D: x ∈ ℝ R: y > −4 b) vertical translation (k) and reflection in the x-axis (a) b) horizontal translation (h) and reflection in the y-axis (c) c) y = 4 b) c) asymp: x = −2 5 x = − ≅ −1.7 3 y = log 3 2 + 1 ≅ 2.6 D: x > −2 R: y ∈ ℝ Page 19 of 19 d) asymp: y = 6 x=− log 2 1 log�3� y=3 D: x ∈ ℝ R: y < 6 ≅ 0.6 asymp: x = −4 x=− 63 16 ≅ −3.9 y = −3 D: x > −4 R: y ∈ ℝ
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Hi Danica. Please see attached file.

WCLN PCMath 12 - Rev. 2019

Name: _______________________________
UNIT 4 - LEARNING GUIDE: Logarithms
Instructions:
Using a pencil, complete the following questions as you work through the related lessons. Show
ALL work as it is explained in the lessons. Do your best and ask your instructor if you do not
understand any questions!

Lesson 1 – Review the Exponent Laws
1. Simplify the following and write answers with positive exponents only.
a)

(x 2 )4
x −3

b)

= x (2∗4 − −3)

= m(−2−3) n(−3−1)

= x11

=

2

x 2∗2 x
= 2 ∗ 3
y
2y

=

g)

x 4+1
x5
=
2y 2+2 2y 5

(xy)2

1
m5 n4

e)

9y 6 2
( 2 ) ÷ xy
4x
1

=

1

42 x
=

1

2−2 x −2
33 x (2∗3)

=

1
1
=
4 ∗ 27 ∗ x (2+6) 108x 8
(3m)−1 n 3
( )
(2n)2 m2

1
n3
=

3 ∗ 22 ∗ mn2 m2∗3

1
1∗
xy
2∗
2

3y 3 1
3y 2

=
2x1 xy 2x 2

x2y2
2x 3
1
=
∗(
) ∗ = 22 x 2+2∗3 ∗ y 2−1
1
1
y

(2x)−2
(3x 2 )3

=

f)

92 y 6∗2

1 −2
( 3) ÷ y
2x
2

c)

1

−1

x2
2y 3
d) ( ) (
)
y
x

(m2 n3 )−1
m3 n

=

1
n3
n

=
2
6
12mn m
12m7
2

h)

(2m2 )−1
m 4
(
)
÷
(
)
n3
3n3

2
1
m4
1
81n12 81n6
= ( 2 3 ) ÷ 4 3∗4 =

=
2m n
3 n
4m4 n6
m4
4m8

= 4x 8 y
Page 1 of 19

WCLN PCMath 12 - Rev. 2019

2. Simplify and then evaluate the expressions for x = -1, y = 2, z = 3.
a) (3x 2 y −1 z 2 ) (4x −1 y 2 z )

= 12x

(2xy 3 z −1 )2
x2y4

b)

22 x 2 y 3∗2 z −1∗2
4x 2 y 6 z −2
=
=
x2 y4
x2y4

2−1 −1+2 2+1

y

z

= 12xyz 3

= 22 x 2−2 y 6−4 z −2 =

= 12(−1)(2)(3)3 = −648

=

4y 2
z2

4(2)2 16
=
(3)2
9

Lesson 2 – Solving Equations Involving Exponents
1. Solve for x using the reciprocal of the exponent.
2

a) x 3 = 4
3
2 2
(x 3 )

= 43/2

x=8

3

40
=
=8
5

2
3 3
(x 2 )

x=4

1
(x −2

x=

c) 5x 2 = 40
3
x2

1

b) x −2 = 9

= 82/3



)

2
1

2

= 9 −1

1
1
=
2
9
81

2

d) (4x + 1)3 = 25

((4x +

3
2 2
1)3 )

4x + 1 = 125

x =

Page 2 of 19

3

= 252

125 − 1
= 31
4

WCLN PCMath 12 - Rev. 2019

2. Solve for x by converting to common bases. Show all your work.
1 x
x
a) 2 = 32
b) ( ) = 81
C) 5x = 625
3
2 =2

1 x
1 −1
1 −1
1 −4
( ) = ( ) = ( 4) = ( )
3
81
3
3

5x = 54

x=5

x = −4

x=4

d) 3x−2 = 27

e) 42x = 8

f) 27x−1 = 9

x

5

3

3x−2 = 33

42x = 42

x−2 =3

2x =

x=5

x=

g) 3(16x+2 ) = 96
16x+2 =

96
= 32
3

3
4

x=

2x

2x − 1 =

5
3
→ x=−
4
4

j) 4x = 8...


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