Exponents & Logarithm
Law of exponents
-1h
am ✗ an = am
am
n
am
-
÷ an __
( am )n = amn
a-
n
= ÷
"
( ab ) = anbn
⇐ In = ÷
a
? =ⁿaT
Law of logarithm Proving
"
109 ax 109 ay toga ✗Y
a = b ⇔ 109bar = ✗ + =
" M
10910 = × let logax =
, 109 ay = N
10109 "
= × am = ×
,
a
"
= y
Ine
"
= x XY = am .
an
-1h
e
'" "
×
= am
=
toga XY = 109AM + logay 10gal ✗ Y ) =
Mt N
109A } = 109 ax -
109 ay toga ✗ + 109A Y =
109A ( XY )
10gal = 0
109 , a
109bar =
109C b
/ 09 ax = ✗
a
, Sequence
Arithmetic Progression Geometric progression
Un Uit (n 1) d Un rn '
-
= -
= u ,
hi ( I - rn )
Sn =
{ [ 241 1- ( n -
1) d ] =
§ ( Ut -14h ) Sn =
, -
rn
Irl < I
Un+ , -
un = Un -
Un -
i = constant un -11 Un
= = constant
un un -
,
Un = Sn -
Sn -
1 ,
n ≥ 2
Ul
so =
Uz = 52 -
Si Us = 55 -
Su , -
r
sequence : a- d. a
,
a + a sequence :
a. ar
,
Sigma
n
Ur total no . Of terms = n -
Mt I
r=m
Rules of summation
sum Difference of sums
of constant
n n
,É=
m-l
a = ata -1 . . .
+ at at a = na I Ur = Eur -
Eur
, r=M r= I 8=1
n times
compound interest Approximation calculation
r 010 = It r=c - i real rate of return
⇒ per
↑ annum
given ◦ to
change
kn
✓
f- ✓ = PV ' +
k( 100 )
"
real value = PV ( it )
real value °
of = - I ✗ 100 /0
real value return pv
Annual calculation
C
I -1 Too
' +
¥
'
io / ◦ =
I +
, Counting principles & Algebra
AND rule OR rule
n (A and B) = n( A) ✗ n( B) n (A or B) = n( A) + n( B)
Combination
Permutation
n ! n !
"
Pr =
ncr =
(n -
r )! r ! ( n -
r) !
order matters ⇒ order matters
⇒ ✗
Algebra
if 1×1<1 I -
K -1×2×3-1 . _ .
= ( I -1×1-1
,
( It >c)
n
= It nc , x th ( 27C
≥
+
n ( 3×3-1 . . . ✗
n
ncn -
1) z ncn 1) ( n 2)
It nx +
×3+
- -
=
× + n
# ✗
. . .
3!
Partial fractions
px -19 A B ☆ Expansions converge for 1×1<1
=
+
( 9kt b) ( (✗ + d) ax + b CK 1- d
pxtq = A ( CK -1 d) + Bcaktb )
✗ = & , y=B,z=8 ⇒ unique solution
0=8 81=0 ⇒ no solution inconsistent consistent
,
0=0 ⇒ infinitely many solutions
Law of exponents
-1h
am ✗ an = am
am
n
am
-
÷ an __
( am )n = amn
a-
n
= ÷
"
( ab ) = anbn
⇐ In = ÷
a
? =ⁿaT
Law of logarithm Proving
"
109 ax 109 ay toga ✗Y
a = b ⇔ 109bar = ✗ + =
" M
10910 = × let logax =
, 109 ay = N
10109 "
= × am = ×
,
a
"
= y
Ine
"
= x XY = am .
an
-1h
e
'" "
×
= am
=
toga XY = 109AM + logay 10gal ✗ Y ) =
Mt N
109A } = 109 ax -
109 ay toga ✗ + 109A Y =
109A ( XY )
10gal = 0
109 , a
109bar =
109C b
/ 09 ax = ✗
a
, Sequence
Arithmetic Progression Geometric progression
Un Uit (n 1) d Un rn '
-
= -
= u ,
hi ( I - rn )
Sn =
{ [ 241 1- ( n -
1) d ] =
§ ( Ut -14h ) Sn =
, -
rn
Irl < I
Un+ , -
un = Un -
Un -
i = constant un -11 Un
= = constant
un un -
,
Un = Sn -
Sn -
1 ,
n ≥ 2
Ul
so =
Uz = 52 -
Si Us = 55 -
Su , -
r
sequence : a- d. a
,
a + a sequence :
a. ar
,
Sigma
n
Ur total no . Of terms = n -
Mt I
r=m
Rules of summation
sum Difference of sums
of constant
n n
,É=
m-l
a = ata -1 . . .
+ at at a = na I Ur = Eur -
Eur
, r=M r= I 8=1
n times
compound interest Approximation calculation
r 010 = It r=c - i real rate of return
⇒ per
↑ annum
given ◦ to
change
kn
✓
f- ✓ = PV ' +
k( 100 )
"
real value = PV ( it )
real value °
of = - I ✗ 100 /0
real value return pv
Annual calculation
C
I -1 Too
' +
¥
'
io / ◦ =
I +
, Counting principles & Algebra
AND rule OR rule
n (A and B) = n( A) ✗ n( B) n (A or B) = n( A) + n( B)
Combination
Permutation
n ! n !
"
Pr =
ncr =
(n -
r )! r ! ( n -
r) !
order matters ⇒ order matters
⇒ ✗
Algebra
if 1×1<1 I -
K -1×2×3-1 . _ .
= ( I -1×1-1
,
( It >c)
n
= It nc , x th ( 27C
≥
+
n ( 3×3-1 . . . ✗
n
ncn -
1) z ncn 1) ( n 2)
It nx +
×3+
- -
=
× + n
# ✗
. . .
3!
Partial fractions
px -19 A B ☆ Expansions converge for 1×1<1
=
+
( 9kt b) ( (✗ + d) ax + b CK 1- d
pxtq = A ( CK -1 d) + Bcaktb )
✗ = & , y=B,z=8 ⇒ unique solution
0=8 81=0 ⇒ no solution inconsistent consistent
,
0=0 ⇒ infinitely many solutions
