Properties of Logarithms
Objective: To learn and apply the
properties of logarithms.
Who is the “Father of Logarithms?
Š John Napier,
mathematician from Scotland
Š “Napier’s Bones”
Š Here’s a look at his burial
site and family pharmacy.
Real-World Connection
Š Logarithms are used in applications
involvingg sound intensityy &
decibel level.
Think about this…
Š If a logarithm is the inverse of an exponential, what
do you think we can surmise about the properties of
logarithms?
Š 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
Š Product Property: logb(MN) = logbM + logbN
The logarithm of a product is the __________
off the
th logarithms
l
ith
off th
the factors.
f t
Š Ex) logbx3
+ logby =
1
Example
Š Express as a single logarithm:
logg 3 x + log
g3 w
2
Example – Product Rule
Š Use the product rule to expand each
logarithmic expression:
Š A) log6(7)(11)
B) log(100x)
QUOTIENT RULE
Š Quotient Property
logb(M/N) = logbM – logbN
The logarithm of a quotient is the logarithm
of the numerator __________ the
logarithm of the denominator.
Ex) log2w
Example
Š Express as a difference of
logarithms.
l a
log
10
b
- log216 =
Example – Quotient Rule
Š Use the quotient rule to expand each logarithmic
expression:
⎛ e5 ⎞
⎛ 23 ⎞
A) log8 ⎜ ⎟
B) ln ⎜ ⎟
⎝ x ⎠
⎝ 11 ⎠
POWER RULE
Š Power Property: logbMp = p logbM
The logarithm of a power of M is the exponent
__________ the logarithm of M.
Š Ex) log2x3 =
2
Example – Power Rule
Example
Š Express as a product.
−3
a
log 7
Extra Practice
Š Express as a product.
5
a
log
11
loga 5 11 = loga 111/5
1
= loga 11
5
Example – Expanding Logs
Š Use log properties to expand each expression as
much as possible.
a)) log
l b ( x4 3 y )
⎛ x ⎞
b) log
l 5 ⎜⎜
3 ⎟
⎟
⎝ 25 y ⎠
Š Use the power rule to expand each logarithmic
expression:
A)) logg 6 39
C)) log(
g( x + 4)) 2
B)) ln 3 x
Expanding Logarithmic Expressions
Š Use properties of logarithms to change one logarithm
into a sum or difference of others.
⎛ 724 x ⎞
⎛ 1⎞
Š Example log
⎟ = log
l ⎜
l
72 + log
l ⎜ x 4 ⎟ − log
l ( y4 )
6
⎜ 4 ⎟
⎝ y ⎠
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
Expanding Logs – Express as a sum
or difference.
w3 y 4
log a 2
z
3
More Practice Expanding
a) log27b
c) log7a3b4
Let’s reverse things.
Š Express as a single logarithm.
log
og w 125
5 − log
og w 25
5
b) log(y/3)2
Pencils down. Watch and listen.
Š Express as a single logarithm.
1
6 logg b x − 2 logg b y + log
gb z
3
Š Solution:
6 log b x − 2 log b y +
Š We can also use the properties of logarithms to
condense expressions or “write as a __________
logarithm”.
g
1
log b z = log b x 6 − log b y 2 + log b z 1 / 3
3
x6
= log b 2 + log b z 1 / 3
y
= log b
x 6 z1 / 3
x6 3 z
, or log b
y2
y2
Examples – Condensing Logs
Š Write as a single logarithm.
a) log 25 + log 4
Condensing Logarithmic Expressions
b) log(7 x + 6) − log x
Write as a single logarithm.
1
a) 2 ln x + ln( x + 5)
3
b) 2 log( x − 3) − log x
4
More Practice
Write as a single logarithm.
c)
1
log b x − 2 log b 5 − 10 log b y
4
Š d) Write 3log2 + log 4 – log 16 as a single
logarithm.
Š e) Can you write 3log29 – log69 as a single
logarithm?
Review Examples
Review of Properties
Š The Logarithm of a Base to a Power
For any base a and any real number x,
logga a x = x.
(The logarithm, base a, of a to a power is the power.)
Š Simplify.
a) loga a 6
b) ln e −8
• 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.)
Simplify.
Š A)
7 log7 w
Š B)
eln 8
Change of Base Formula
Š 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.
Š For ease of computing a logarithm, we may want to
change from one base to another using the formula
log b M =
log M
log b
or log b M =
ln M
ln b
5
Summary of
Properties of Logarithms
Change of Base Examples
Š Use common logs to evaluate log7 2506.
Š Use natural logs to evaluate log7 2506.
For a > 0, a ≠ 1,and any real num ber k,
1 ) lo g
a
a = 1, l n e = 1
2 ) lo g a 1 = 0 ,
ln 1 = 0
A d d i t io n a l L o g a r it h m ic P r o p e r t ie s
3 ) lo g a a k = k
4 ) a lo g a 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
loga
7) Power Rule
x
= loga x-loga y
y
loga xr = rloga x
6