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!