Math Calculations using powers

This document provides examples of simplifying calculations using powers in Math. It includes step-by-step solutions for various problems involving powers. The document also covers rearranging equations and solving logarithmic equations. The subject is Math.

Running head: Math 1
Math
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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