Unit V: Properties of Logarithms
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Transcript Unit V: Properties of Logarithms
Properties of Logarithms
Section 4.3
JMerrill, 2005
Revised, 2008
Rules of Logarithms
If M and N are positive real numbers and b is ≠ 1:
The Product Rule:
logbMN = logbM + logbN
(The logarithm of a product is the sum of the logarithms)
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:
The Product Rule:
logbMN = logbM + logbN
(The logarithm of a product is the sum of the logarithms)
Example: log4(7 • 9) = log47 + log49
Example: log (10x) = log10 + log x
You do: log8(13 • 9) = log813 + log89
You do: log7(1000x) = log71000 + log7x
Rules of Logarithms
If M and N are positive real numbers and b ≠ 1:
The Quotient Rule
M
logb
N
log b M log b N
(The logarithm of a quotient is the difference of the logs)
Example:
x
log log x log 2
2
Rules of Logarithms
If M and N are positive real numbers and b ≠ 1:
The Quotient Rule
M
logb log b M log b N
N
(The logarithm of a quotient is the difference of the logs)
x
log
log
x
log
2
Example:
2
14
log 7
x
You do:
log7 14 log7 x
Rules of Logarithms
If M and N are positive real numbers, b ≠ 1, and p is
any real number:
The Power Rule:
logbMp = p logbM
(The log of a number with an exponent is the product of the
exponent and the log of that number)
Example: log x2 = 2 log x
Example: ln 74 = 4 ln 7
You do: log359 = 9log135
Challenge: ln x ln x 2 1 ln x
2
Prerequisite to Solving Equations
with Logarithms
Simplifying
Expanding
Condensing
Simplifying (using Properties)
log94 + log96 = log9(4 • 6) = log924
log 146 = 6log 14
3
log 3 log 2 log
2
You try: log1636 - log1612 = log163
You try: log316 + log24 = Impossible!
You try: log 45 - 2 log 3 = log 5
Using Properties to Expand
Logarithmic Expressions
Expand: log b x 2 y
log b
1
2 2
x y
log b x log b
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
log 6
4
36
y
log 6
1
x3
36 y 4
log 6
1
x3
log 6 36 y 4
log 6
1
x3
log 6 36 log 6 y 4
1
log 6 x log 6 36 4 log 6 y
3
1
log 6 x 2 4 log 6 y
3
Condensing
Condense:
logb M logb N 3logb P
logb MN 3log b P
logb MN logb P
MN
logb 3
P
Product Rule
3
Power Rule
Quotient Rule
Condensing
Condense:
1
logb M logb N logb P
2
logb M logb N logb P
log b
1
2
1
2
MN
MN
or log b
P
P
Bases
Everything we do is in Base 10.
We count by 10’s then start over. We change our
numbering every 10 units.
In the past, other bases were used.
In base 5, for example, we count by 5’s and change
our numbering every 5 units.
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
Examine the following problems:
log464 = x
we know that x = 3 because 43 = 64, and the base of
this logarithm is 4
log 100 = x
– If no base is written, it is assumed to be base 10
We know that x = 2 because 102 = 100
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
.
Example log58 =
log 5
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!