Power Rules and Logarithm Rules
Definition
Definition
a times itself n times =
Examples
The power to which a number has to be
raised
n
a
a⋅a = a
a⋅a⋅a = a
2
log a ( an ) = n
3
Examples
log 10 (100) = 2
7
a⋅a⋅a⋅a⋅a⋅a⋅a = a
log 3 (27) = 3
Reciprocal Rule
1
= a−x
ax
3
1/5 = 5
−3
n
−n
log a ( 1/a ) = log a ( a
log 5 (
) = −n
1
) = −3
125
Product Rule
x
a ⋅a
y
2
( x+ y)
= a
3
5
2 ⋅2 = 2
x
y
log a ( a ⋅a ) = x + y
2
3
log 2 (2 ⋅2 ) = 5
4⋅8 = 32
Quotient Rule
a
x
ay
( x− y)
= a
54
(4−2)
2
= 5
= 5
2
5
log (
a
x
ay
log 5 (
) = x− y
54
) = 2
2
5
625
= 25
25
Power and Logarithm Rules
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Power Rule
(a x ) y = a xy
log a ( (a x ) y ) = xy
(32 )3 = 36
log 3 ( (32 )3 ) = 6
Product Distribution Rule
x
log c (ab) = log c a + log c b
x
x
log c ((ab) ) = x (log c a + log c b)
(3⋅5)2 = 32⋅5 2
log c ((3⋅5)2) = 2(log c 3 + log c 5)
(ab) = a ⋅b
Quotient Distribution Rule
a
log c ( ) = log c a − log c b
b
a x
ax
( ) =
b
bx
3
x
3
2
2
( ) = 3
5
5
Power Distrubtion Rule
a x
log c ( ( ) ) = x (log c a − log c b)
b
2 3
log ( ( ) ) = 3( log 2 − log 5)
5
b
log c a = b log c a
log 2 53 = 3 log 2 5
Identity Rule
a
0
= 1
Power and Logarithm Rules
log 1 = 0
a
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Base Reciprocal Rule
log (1 / a) b = − log a b
log (1 /2 ) 8 = − log 2 8 = −3
Radical Rule
(1/n)
a
125
(1 /3)
=
n
n
√a
log a ( √a) = 1 /n
√3 125 = 5
log 3 ( √3) = 1/ 2
=
Decay Rule (half-life)
(t /h)
A = A 0 (1/2)
Where A is the current quantity
A0 is the starting quantity
t is the time that has passed
h is the half life
log A = log A 0 − (t / h) log( 2)
log 2 A = log 2 A0 − t / h
Growth Rule (doubling time)
(t /d)
A = A 0 (2)
where A is the current quantity
A0 is the starting quantity
t is the time that has passed
d is the doubling time
Power and Logarithm Rules
log A = log A 0 + ( t/d ) log 2
log 2 A = log 2 A0 + (t / d )
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Change of Base
log x (b)⋅ log b (a) = log x (a)
log b a =
log 16 1024 =
log 3 16 =
log x a
log x b
log 2 1024
10
=
= 2.5
log2 16
4
log 2 16
4
=
= 2.523 ...
log 2 3
1.585 ...
Compound Interest
A = P⋅(1+i)n
PV =
Where
A
n
(1+i)
A is the final amount
P is the principal
PV is the Present Value
i is the interest rate
n is the number of interest periods
Power and Logarithm Rules
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Annuity
A =
R =
Where
R⋅[(1+i)n − 1]
i
Ai
[( 1+i )n − 1]
A is the final amount
R is the regular payment
i is the interest rate
n is the number of payments
Power and Logarithm Rules
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