Lesson 3.4
Properties of Logarithms
Objective: To learn and apply the
properties of logarithms.
Think about this…
 If a logarithm is the inverse of an exponential, what
do you think we can surmise about the properties of
logarithms?
 They should be the inverse of the properties of
exponents! For example, if we add exponents when
we multiply in the same base, what would we do to
logs when they are being multiplied?
PRODUCT RULE
 Product Property: logb(MN) = logbM + logbN
The logarithm of a product is the sum of the
logarithms of the factors.
 Ex) logbx3
+ logby =
See Example 1
 Express as a single logarithm:
log3 x  log3 w
2
Check Point 1
 Use the product rule to expand each
logarithmic expression:
 A) log6(7  11) B) log(100x)
QUOTIENT RULE
 Quotient Property
logb(M/N) = logbM – logbN
The logarithm of a quotient is the logarithm
of the numerator minus the logarithm of
the denominator.
Ex) log2w
- log216 =
See Example 2
 Express as a difference of
logarithms.
10
log a
b
Check Point 2
 Use the quotient rule to expand each logarithmic
expression:
5


23
e
 
A) log8  
B) ln  
 x 
 11 
POWER RULE
 Power Property: logbMp = p logbM
The logarithm of a power of M is the exponent
times the logarithm of M.
 Ex) log2x3 =
See Example 3
 Express as a product.
3
a
log 7
Check Point 3
 Use the power rule to expand each logarithmic
expression:
9
A) log6 3
3
B) ln x
C) log( x  4)
2
Extra Practice
 Express as a product.
5
log a 11
log a 11  log a 11
5
1/ 5
1
 log a 11
5
Expanding Logarithmic Expressions
(See blue box on page 416.)
 Use properties of logarithms to change one logarithm
into a sum or difference of others.
1
4




72
x
 Example log 
4
  log 72  log  x   log ( y 4 )
6
 y4 


6
6




6
1
 log 6 (36  2)  log 6 ( x)  4 log 6 ( y )
4
1
2
 log 6 (6 )  log 6 (2)  log 6 ( x)  4 log 6 ( y )
4
1
 2  log 6 (2)  log 6 ( x)  4 log 6 ( y )
4
See Example 4
 Check Point 4: Use log properties to expand each
expression as much as possible.
a) log b ( x
43
y)
 x 
b) log 5 
3 

25
y


Expanding Logs – Express as a sum
or difference.
3
wy
log a 2
z
4
More Practice Expanding
a) log27b
b) log(y/3)2
c) log7a3b4
Condensing Logarithmic Expressions
 We can also use the properties of logarithms to
condense expressions or “write as a single logarithm”.
Let’s reverse things.
 Express as a single logarithm.
log w 125  log w 25
Pencils down. Watch and listen.
 Express as a single logarithm.
1
6log b x  2log b y  log b z
3
 Solution:
1
6log b x  2log b y  log b z  log b x 6  log b y 2  log b z1/ 3
3
x6
 log b 2  log b z1/ 3
y
x 6 z1/ 3
x6 3 z
 log b
, or log b
2
y
y2
Check Point 5
 Write as a single logarithm.
a) log 25  log 4
b) log(7 x  6)  log x
Check Point 6
Write as a single logarithm.
1
a) 2 ln x  ln( x  5)
3
b) 2 log( x  3)  log x
Check Point 6
Write as a single logarithm.
1
c) log b x  2 log b 5  10 log b y
4
More Practice
 d) Write 3log2 + log 4 – log 16 as a single
logarithm.
 e) Can you write 3log29 – log69 as a single
logarithm?
Review of Properties
(from Lesson 3.2)
 The Logarithm of a Base to a Power
For any base a and any real number x,
loga a x = x.
(The logarithm, base a, of a to a power is the power.)
• A Base to a Logarithmic Power
For any base a and any positive real number x,
a
log a x
 x.
(The number a raised to the power loga x is x.)
Examples
 Simplify.
a) loga a 6
b) ln e 8
Simplify.
 A)
 B)
7
log 7 w
e
ln 8
Change of Base Formula
 The 2 bases we are most able to calculate logarithms
for are base 10 and base e. These are the only bases
that our calculators have buttons for.
 For ease of computing a logarithm, we may want to
switch from one base to another using the formula
log M
logb M 
log b
ln M
or log b M 
ln b
See Examples
 Use common logs to evaluate log7 2506.
 Use natural logs to evaluate log7 2506.
Solve:

Summary of
Properties of Logarithms
For a > 0, a  1,and any real number k,
1) log a  1, ln e = 1
a
2) loga 1 = 0, ln 1 = 0
Additional Logarithmic Properties
3) loga a k = k
4) aloga k = k, k > 0
Summary of
Properties of Logarithms (cont.)
For x > 0, y  0, a  0, a  1,and any real number r,
5) Pr oduct Rule
loga xy = loga x +loga y
6) Quotient Rule
x
loga  loga x -loga y
y
7) Power Rule
loga x r  r loga x