Properties of Logarithms
Section 3-3
Rules of Logarithms
If M and N are positive real numbers and b is ≠ 1:
ƒ
The Product Rule:
ƒ
logbMN = logbM + logbN
(The logarithm of a product is the sum of the logarithms)
ƒ
ƒ
Example: log4(7 • 9) = log47 + log49
Example: log (10x) = log10 + log x
Rules of Logarithms
If M and N are positive real numbers and b ≠ 1:
ƒ
The Product Rule:
ƒ
logbMN = logbM + logbN
(The logarithm of a product is the sum of the logarithms)
ƒ
ƒ
ƒ
ƒ
Example: log4(7 • 9) = log47 + log49
Example: log (10x) = log10 + log x
You do: log8(13 • 9) =
You do: log7(1000x) =
Rules of Logarithms
If M and N are positive real numbers and b ≠ 1:
ƒ
The Quotient Rule
⎛M
log b ⎜
⎝N
⎞
⎟ = log b M − log b N
⎠
(The logarithm of a quotient is the difference of the logs)
ƒ
Example:
⎛ x⎞
lo g ⎜ ⎟ = lo g x − lo g 2
⎝2⎠
Rules of Logarithms
If M and N are positive real numbers and b ≠ 1:
ƒ
The Quotient Rule
⎛M ⎞
log b ⎜ ⎟ = log b M − log b N
⎝N⎠
(The logarithm of a quotient is the difference of the logs)
ƒ
⎛ x⎞
lo
g
=
lo
g
x
−
lo
g
2
⎜
⎟
Example:
⎝2⎠
ƒ
⎛ 14 ⎞
log 7 ⎜ ⎟ =
⎝ x⎠
You do:
log 7 14 − log 7 x
Rules of Logarithms
If M and N are positive real numbers, b ≠ 1, and p is
any real number:
ƒ The Power Rule:
ƒ logbMp = p logbM
(The log of a number with an exponent is the product of the
exponent and the log of that number)
ƒ
ƒ
ƒ
ƒ
Example: log x2 = 2 log x
Example: ln 74 = 4 ln 7
You do: log359 =
Challenge: ln x
Prerequisite to Solving Equations
with Logarithms
ƒ
ƒ
ƒ
Simplifying
Expanding
Condensing
Simplifying (using Properties)
ƒ log94 + log96 = log9(4 • 6) = log924
ƒ log 146 = 6log 14
3
log3 − log2 = log
2
ƒ
ƒ
ƒ
You try:
You try:
You try:
log1636 - log1612 =
log316 + log24 =
log 45 - 2 log 3 =
Using Properties to Expand
Logarithmic Expressions
ƒ Expand: log x 2 y
b
log b
1
2 2
x y
log b x + logb
2
Use exponential notation
1
y2
Use the product rule
1
2 log b x + log b y Use the power rule
2
Expanding
⎛ 3 x
lo g 6 ⎜⎜
4
y
3
6
⎝
⎞
⎟⎟
⎠
1
3
lo g 6
x
36 y 4
lo g 6 x
lo g 6 x
1
3
1
3
− lo g 6 3 6 y 4
(
− lo g 6 3 6 + lo g 6 y 4
1
lo g 6 x − lo g 6 3 6 − 4 lo g 6 y
3
1
lo g 6 x − 2 − 4 lo g 6 y
3
)
Condensing
ƒ Condense:
log b M + log b N − 3log b P
log b MN − 3log b P
log b MN − logb P
MN
log b 3
P
Product Rule
3
Power Rule
Quotient Rule
Condensing
ƒ Condense:
1
( logb M + logb N − logb P )
2
( logb M + logb N − logb P )
log b
1
2
1
⎞2
MN
⎛ MN
or logb ⎜
⎟
P
⎝ P ⎠
Bases
ƒ Everything we do is in Base 10.
ƒ We count by 10’s then start over. We change our
numbering every 10 units.
ƒ In the past, other bases were used.
ƒ In base 5, for example, we count by 5’s and change
our numbering every 5 units.
ƒ We don’t really use other bases anymore,
but since logs are often written in other
bases, we must change to base 10 in order
to use our calculators.
Change of Base
ƒ Examine the following problems:
ƒ log464 = x
ƒ we know that x = 3 because 43 = 64, and the base of
this logarithm is 4
ƒ log 100 = x
– If no base is written, it is assumed to be base 10
ƒ We know that x = 2 because 102 = 100
ƒ But because calculators are written in base
10, we must change the base to base 10 in
order to use them.
Change of Base Formula
logM
logb M =
logb
log 8
= 12900
.
ƒ Example log58 =
log 5
ƒ This is also how you graph in another base.
Enter y1=log(8)/log(5). Remember, you don’t
have to enter the base when you’re in base 10!