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Core Connections Integrated III, 2015
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Core Connections Integrated III, 2015 View details
1. Section 9.1
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Exercise 51 Page 448

Practice makes perfect
a Wewant to solve the following equation.

7 = 4.2^xWe will do so by applying inverse operations to isolate x. In particular, we will use the Properties of Logarithms to eliminate the exponent from the equation.

7 = 4.2^x

log(LHS)=log(RHS)

log 7= log 4.2^x

log(a^m)= m*log(a)

log 7= x log 4.2
DivEqn

.LHS /log 4.2.=.RHS /log 4.2.

log 7/log 4.2 = x
RearrangeEqn

Rearrange equation

x = log 7/log 4.2
UseCalc

Use a calculator

x ≈ 1.35595507774...
RoundDec

Round to 3 decimal place(s)

x ≈ 1.356

To check our solution, we will substitute x ≈ 1.356 into the original equation. Note that since the answer is approximate, we will check whether, after substitution, the left-hand side approximately equals the right-hand side.

7 = 4.2^x

x ≈ 1.356

7 ? ≈ 4.2^(1.356)
UseCalc

Use a calculator

7 ≈ 7.00045128521... ✓

Therefore, x ≈ 1.356 is a good approximation of the solution.

b We want to solve the following equation.

3x^5 = 126Let's isolate x by applying inverse operations to both sides of the equation.

3x^5 = 126
DivEqn

.LHS /3.=.RHS /3.

x^5 = 42
RadicalEqn

sqrt(LHS)=sqrt(RHS)

x = sqrt(42)
UseCalc

Use a calculator

x = 2.111785...
RoundDec

Round to 3 decimal place(s)

x ≈ 2.112

To check our solution, we will substitute x ≈ 2.112 into the original equation. Note that since the answer is approximate, we will check whether, after substitution, the left-hand side approximately equals the right-hand side.

14 = 2 * 4^x - 10

x ≈ 2.112

3( 2.112)^5 ? ≈ 126
CalcPow

Calculate power

3(42.021308...)? ≈ 126
UseCalc

Use a calculator

126.063924... ≈ 126 ✓

Therefore, x ≈ 2.112 is a good approximation of the solution.

c Wewant to solve the following equation.

14= 2 * 4^x -10 Let's first use inverse operations to isolate 4^x.

14= 2 * 4^x -10
AddEqn

LHS+10=RHS+10

24 = 2* 4^x
DivEqn

.LHS /2.=.RHS /2.

12 = 4^x

Now that 4^x is isolated on one hand of our equation, let's continue solving it using the Properties of Logarithms.

12 = 4^x

log(LHS)=log(RHS)

log 12= log 4^x

log(a^m)= m*log(a)

log 12= x log 4
DivEqn

.LHS /log 4.=.RHS /log 4.

log 12/log 4 = x
RearrangeEqn

Rearrange equation

x = log 12/log 4
UseCalc

Use a calculator

x ≈ 1.792481...
RoundDec

Round to 3 decimal place(s)

x ≈ 1.792

To check our solution, we will substitute x ≈ 1.792 into the original equation. Note that since the answer is approximate, we will check whether, after substitution, the left-hand side approximately equals the right-hand side.

14 = 2 * 4^x - 10

x ≈ 1.792

14 ? ≈ 2 * 4^(1.792) - 10
UseCalc

Use a calculator

14 ≈ 13.983993... ✓

Therefore, x ≈ 1.792 is a good approximation of the solution.

Subchapter links expand_more
arrow_right 9.1.1 Introduction to Periodic Models p.433-434
3
Introduction to Periodic Models 4 (Page 433)
Introduction to Periodic Models 5 (Page 433)
Introduction to Periodic Models 6 (Page 433)
Introduction to Periodic Models 7 (Page 433)
Introduction to Periodic Models 8 (Page 434)
9
arrow_right 9.1.2 Graphing the Sine Function p.438-440
Graphing the Sine Function 13 (Page 438)
Graphing the Sine Function 14 (Page 438)
Graphing the Sine Function 15 (Page 438)
Graphing the Sine Function 16 (Page 438)
Graphing the Sine Function 17 (Page 438)
18
Graphing the Sine Function 19 (Page 439)
Graphing the Sine Function 20 (Page 439)
Graphing the Sine Function 21 (Page 439)
Graphing the Sine Function 22 (Page 439)
Graphing the Sine Function 23 (Page 440)
24
Graphing the Sine Function 25 (Page 440)
Graphing the Sine Function 26 (Page 440)
arrow_right 9.1.3 Unit Circle ↔ Graph p.442-444
Unit Circle ↔ Graph 30 (Page 442)
Unit Circle ↔ Graph 31 (Page 443)
Unit Circle ↔ Graph 32 (Page 443)
Unit Circle ↔ Graph 33 (Page 443)
Unit Circle ↔ Graph 34 (Page 443)
Unit Circle ↔ Graph 35 (Page 443)
Unit Circle ↔ Graph 36 (Page 444)
arrow_right 9.1.4 Graphing and Interpreting the Cosine Function p.447-449
Graphing and Interpreting the Cosine Function 45 (Page 447)
Graphing and Interpreting the Cosine Function 46 (Page 447)
Graphing and Interpreting the Cosine Function 47 (Page 447)
Graphing and Interpreting the Cosine Function 48 (Page 447)
Graphing and Interpreting the Cosine Function 49 (Page 447)
Graphing and Interpreting the Cosine Function 50 (Page 448)
Graphing and Interpreting the Cosine Function 51 (Page 448)
Graphing and Interpreting the Cosine Function 52 (Page 448)
Graphing and Interpreting the Cosine Function 53 (Page 448)
Graphing and Interpreting the Cosine Function 54 (Page 448)
Graphing and Interpreting the Cosine Function 55 (Page 449)
Graphing and Interpreting the Cosine Function 56 (Page 449)
Graphing and Interpreting the Cosine Function 57 (Page 449)
Graphing and Interpreting the Cosine Function 58 (Page 449)
arrow_right 9.1.5 Defining a Radian p.452-453
Defining a Radian 65 (Page 452)
Defining a Radian 66 (Page 452)
Defining a Radian 67 (Page 452)
Defining a Radian 68 (Page 452)
Defining a Radian 69 (Page 452)
Defining a Radian 70 (Page 453)
71
arrow_right 9.1.6 Building a Unit Circle p.456-457
Building a Unit Circle 76 (Page 456)
Building a Unit Circle 77 (Page 456)
Building a Unit Circle 78 (Page 456)
Building a Unit Circle 79 (Page 456)
Building a Unit Circle 80 (Page 456)
Building a Unit Circle 81 (Page 457)
Building a Unit Circle 82 (Page 457)
arrow_right 9.1.7 The Tangent Function p.460-461
The Tangent Function 88 (Page 460)
The Tangent Function 89 (Page 461)
The Tangent Function 90 (Page 461)
The Tangent Function 91 (Page 461)
The Tangent Function 92 (Page 461)
The Tangent Function 93 (Page 461)
The Tangent Function 94 (Page 461)