logo

MAT4MDS Assignment 2, 2019

   

Added on  2023-01-19

6 Pages787 Words97 Views
 | 
 | 
 | 
Order Id 953574 MAT4MDS ASSIGNMENT 2, 2019
Question 1
a) Solving these equations for x
i) loge(loge(3x)) = 0 applying the log rule 0 = ln(e0) = ln(1)
loge(loge(3x)) = ln(1)
loge(3x) = 1
e1 = 3x ⟹ 𝐱 = 𝐞𝟏
𝟑
ii) ex2−5x+6 = 1
ln(ex2−5x+6) = ln(1)
x2 − 5x + 6 = 0 solving the quadratic equation
(x − 3)(x − 2) = 0
∴ 𝐱 = 𝟑 , 𝐱 = 𝟐
iii) log3(x2 − 9) − log3(x + 3) = 2 − log3(x + 1)
log3(x2 − 9) − log3(x + 3) + log3(x + 1) = 2
log3 ((x2 − 9)(x + 1)
(x + 3) ) = 2
32 = (x + 3)(x − 3)(x + 1)
(x + 3) ⟹ (x − 3)(x + 1) = 9
x2 − 2x − 12 = 0
∴ 𝐱 = 𝟏 + √𝟏𝟑 𝐎𝐫 𝐱 = 𝟏 + √𝟏𝟑
𝐱 = 𝟒. 𝟔𝟎𝟓𝟓𝟓𝟏𝟐𝟕𝟓 𝐨𝐫 𝐱 = −𝟐. 𝟔𝟎𝟓𝟓𝟓𝟏𝟐𝟕𝟓 in decimal format
iv) 9𝑥 − 2.3𝑥+1 = 7
MAT4MDS Assignment 2, 2019_1

Using trial and error method
𝒙 = 𝟏. 𝟏𝟕𝟎𝟖𝟑
b) Proof
i) loga(b) logb(a) = 1
logb(a) = 1
loga b
1 = 1
loga b × 1
logb a × (loga(b) logb(a))
1 = ( 1
loga b × loga(b)) × ( 1
logb a × logb(a))
1 = 1 × 1 = 1
ii) log1
b
(x) = − logb(x)
logb−1 (x) = − logb(x) use of change of base rule
ln(x)
ln(b−1) = − ln(x)
ln(b)
ln(x)
−ln(b) = − ln(x)
ln(b)
ln(x)
ln(b) = − ln(x)
ln(b)
c) Graphing the following equations on the same axis, c=8
𝑦 = 𝑥, (1
5)
𝑥
, (1
8)
𝑥
MAT4MDS Assignment 2, 2019_2

d) Estimating the slope of the red line
The graph is logarithmic on y-axis
s = ∆y
∆x = ∆W
∆n at (1,1) and (2,10)
Where W total number of websites, and n the number of year
s= ∆W
∆n = 1
2−1 = 1 ∆W = 1 for one log cycle
∴ s = 1
yintercept = −10
Thus, y = −10(10)x
The relationship between W and n is Exponential relationship
Question 2
F: [0, 1} → ℝ F(x) = 1 − (1 − xa)b
MAT4MDS Assignment 2, 2019_3

End of preview

Want to access all the pages? Upload your documents or become a member.

Related Documents
Mathematics Assignment
|12
|1256
|68

Mathematics Assignment
|15
|3314
|129

MAT4MDS Assignment 2, 2019
|4
|788
|24

Expand and simplify expressions
|6
|1031
|264

Mathematics Problems with Solutions | Desklib
|8
|714
|256

Maths Integration
|14
|1224
|49