Math 2 Assignment: Simplifying Expressions and Solving Equations

This document contains the solutions to a Math 2 homework assignment. The assignment covers several key mathematical concepts, including simplifying expressions involving powers and exponents, applying logarithmic laws to rewrite and solve logarithmic equations, and rearranging formulas to solve for specific variables. It also includes problems related to exponential functions, such as determining hydrogen ion concentration from pH values and modeling population growth using exponential equations. The solutions are presented with step-by-step calculations and explanations, making it a valuable resource for students studying these topics.

Running head: Math 1
Math
Professor’s Name:
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Date:

MATH 2
Calculations using powers
7. Simplify the following:
a) 1012 × 102 × 10−10=10( 12+2−10 )=104
b) 1023
1015 ×1012 =10 ( 23− (15+12 ) )=10−4
c) 1
10−5 =10 ( 0− (−5 ) ) =105
d) 1014
102 ( 103 ×10−5 )
¿ 10 ( 14 −2 ) ( 10 ( 3 + (−5 ) ) )
¿ 1012 × 10−2
¿ 1010
e) 1016 ×10−20
1012 ×102
¿ 10 (16−20 )
10 (12+ 2 )
¿ 10−4
1014
¿ 10(−4−14 )
¿ 10−18
f) ( 105 )6
=10(5 × 6)=1030
g) 3.67 ×104 ×2.91 ×10−3
33× 6.89 ×1012 =3.67 × 2.91× 10 ( 4 −3 )
33 ×6.89 ×1012
¿ 10.6797× 101
227.37 ×1012
¿ 0.04697 ×10 (1−12 )

MATH 3
¿ 0.04697 ×10−11
¿ 4.697 × 10−13
h) 7.98 ×108 + 4.30 ×106
¿ 7.98 ×108 +0.0430 ×108
¿ ( 7.98+0.0430 ) ×108
¿ 8.023 ×108
Rewrite these as single logarithms by applying the log laws
8. loga x +loga y−loga z=loga ( xy
z )
9. logd ab+logd bc−logd ca=loga ( ab ∙ bc
ca )=logd (b¿ ¿2)=2 logd b ¿
10. log s l4 m2 +log s n−log s lmn=logs ( l4 m2 n
lmn )=logs ( l3 m )
11. logh
x
y + logh
y
z −logh
z
y =logh
( x
y ∙ y
z
z
y )=logh ( x
y ∙ y
z ∙ y
z )=logh ( xy
z2 )
Firstly rewrite these as a single logarithm by applying the log laws. Then solve for x
12. log3 x2+ log3 5−log3 x=log3 15
log3 x2−log3 x=log3 15−log3 5
log3
x2
x =log3
15
5
log3 x=+ log3 3
x=3
13. log7 x +log7 3−log7 5=log7 2

MATH 4
log7 x=log7 2−log7 3+log7 5
log7 x=log7 (2 ×5
3 )
log7 x=log7
10
3
x= 10
3
Week 9
1. Rearrange to make the pronumeral in [] the subject
a) ∆ G=−RT ln ( K ) [K]
∆G
−RT =ln ( K )
K=e
(−∆ G
RT )
b) W =R−T ln ( Q ) [Q]
W −R=−T ln ( Q )
T lnQ=R−W
ln (Q)= R−W
T
Q=e
( R −W
T )
c) A=A0 e5 t [t]
A
A0
=e5 t
ln ( A
A0 )=ln e5 t
ln ( A
A0 )=5 t

MATH 5
t=
ln ( A
A0 )
5
d) N= 5
1+ A e−B [B]
N ( 1+ A e−B ) =5
1+ A e−B = 5
N
A e−B = 5
N −1=5−N
N
e−B = 5−N
NA
ln e− B=ln 5−N
NA
−B=ln ( 5−N
NA )
B=−ln ( 5−N
NA )=ln ( NA
5−N )
2. The pH of a solution is 6, find the hydrogen ion concentration of the solution
pH =−log ¿ ¿
¿
The hydrogen ion concentration =10−6
3. y= A ekt
A= 50, t=20, y=50000
50000=50 e20 k
50000
50 =e20k
ln 50000
50 =ln e20 k

MATH 6
ln 1000=20 k
k = ln 1000
20 =0.3453
k =0.35
4. N= 5 00
1+14 e−0.2 t
a) How many people were initially infected?
t=0
500
1+ 14 e−0.2(0) = 500
15 =33.33
33 people were initially affected
b) How many days will it take for 300 people to be infected?
N = 300
300= 500
1+14 e−0.2 t
300 ( 1+14 e−0.2 t )=500
500
300 =1+14 e−0.2 t
5
3 −1=14 e−0.2t
2
3 =14 e−0.2 t
2
42 =e−0.2t
ln 2
42 =lne−0.2t
−3.0445=−0.2t
t=−3.0445
−0.2 =15.22

MATH 7
t=15 .22 days