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# Lesson 12-3 Dividing Rational Expressions

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```Lesson 3.3, page 400
Properties of Logarithms
Objective: To learn and apply the
properties of logarithms.
Real-World Connection
пЃ·Logarithms are used in applications
involving sound intensity &
decibel level.
пЃ· If a logarithm is the inverse of an exponential, what
do you think we can surmise about the properties of
logarithms?
пЃ· 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, page 400
пЃ· Product Property: logb(MN) = logbM + logbN
The logarithm of a product is the sum of the
logarithms of the factors.
пЃ· Ex) logbx3
+ logby =
See Example 1, pg. 401
пЃ· Express as a single logarithm:
lo g 3 x пЂ« lo g 3 w
2
Check Point 1
пЃ· Use the product rule to expand each
logarithmic expression:
пЃ· A) log6(7 п‚· 11) B) log(100x)
QUOTIENT RULE, page 401
пЃ· Quotient Property
logb(M/N) = logbM вЂ“ logbN
The logarithm of a quotient is the logarithm
of the numerator minus the logarithm of
the denominator.
Ex) log2w
- log216 =
See Example 2, page 402.
пЃ· Express as a difference of
logarithms.
lo g a
10
b
Check Point 2
пЃ· Use the quotient rule to expand each logarithmic
expression:
5
пѓ¦ 23 пѓ¶
A ) log 8 пѓ§
пѓ·
пѓЁ x пѓё
пѓ¦e пѓ¶
B ) ln пѓ§
пѓ·
пѓЁ 11 пѓё
POWER RULE, page 402
пЃ· Power Property: logbMp = p logbM
The logarithm of a power of M is the exponent
times the logarithm of M.
пЃ· Ex) log2x3 =
See Example 3, page 403.
пЃ· Express as a product.
lo g a 7
пЂ­3
Check Point 3
пЃ· Use the power rule to expand each logarithmic
expression:
A ) lo g 6 3
9
B ) ln
3
x
C ) lo g ( x пЂ« 4 )
2
Extra Practice
пЃ· Express as a product.
5
log a 11
log a 11 пЂЅ log a 11
1/5
5
пЂЅ
1
5
log a 11
Expanding Logarithmic Expressions
(See blue box on page 403.)
пЃ· Use properties of logarithms to change one logarithm
into a sum or difference of others.
1
4
пѓ¦
пѓ¶
пѓ¦
пѓ¶
72
x
пЃ· Example log пѓ§
пѓ· пЂЅ log 72 пЂ« log пѓ§ x 4 пѓ· пЂ­ log ( y 4 )
6
пѓ§
пѓЁ
y
4
пѓ·
пѓё
пЂЅ log 6 ( 36 пѓ— 2 ) пЂ«
6
1
4
пЂЅ 2 пЂ« log 6 ( 2 ) пЂ«
1
4
пѓ§
пѓЁ
пѓ·
пѓё
6
log 6 ( x ) пЂ­ 4 log 6 ( y )
пЂЅ log 6 ( 6 ) пЂ« log 6 ( 2 ) пЂ«
2
6
1
4
log 6 ( x ) пЂ­ 4 log 6 ( y )
log 6 ( x ) пЂ­ 4 log 6 ( y )
See Example 4, page 404
пЃ· Check Point 4: Use log properties to expand each
expression as much as possible.
пѓ¦
x пѓ¶
4 3
a) log b ( x
y)
b) log 5 пѓ§
пѓ§ 25 y 3 пѓ·пѓ·
пѓЁ
пѓё
Expanding Logs вЂ“ Express as a sum
or difference.
3
log a
w y
z
2
4
More Practice Expanding
a) log27b
b) log(y/3)2
c) log7a3b4
Condensing Logarithmic Expressions
(See blue box on page 404.)
пЃ· We can also use the properties of logarithms to
condense expressions or вЂњwrite as a single logarithmвЂќ.
пЃ· See Example 5, page 404.
LetвЂ™s reverse things.
пЃ· Express as a single logarithm.
lo g w 1 2 5 пЂ­ lo g w 2 5
Pencils down. Watch and listen.
пЃ· Express as a single logarithm.
6 log b x пЂ­ 2 log b y пЂ«
пЃ· Solution:
6 lo g b x пЂ­ 2 lo g b y пЂ«
1
3
1
3
log b z
lo g b z пЂЅ lo g b x пЂ­ lo g b y пЂ« lo g b z
6
2
6
пЂЅ lo g b
x
y
2
пЂ« lo g b z
6
1/ 3
пЂЅ lo g b
x z
y
2
1/ 3
1/ 3
, o r lo g b
x
6 3
y
2
z
Check Point 5
пЃ· Write as a single logarithm.
a) log 25 пЂ« log 4
b) log(7 x пЂ« 6) пЂ­ log x
Check Point 6
Write as a single logarithm.
a) 2 ln x пЂ«
1
3
ln( x пЂ« 5)
b) 2 log( x пЂ­ 3) пЂ­ log x
Check Point 6
Write as a single logarithm.
c)
1
4
log b x пЂ­ 2 log b 5 пЂ­ 10 log b y
More Practice
пЃ· d) Write 3log2 + log 4 вЂ“ log 16 as a single
logarithm.
пЃ· e) Can you write 3log29 вЂ“ log69 as a single
logarithm?
Review of Properties
(from Lesson 3.2)
пЃ· 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.)
вЂў A Base to a Logarithmic Power
For any base a and any positive real number x,
a
log a x
пЂЅ x.
(The number a raised to the power loga x is x.)
Examples
пЃ· Simplify.
a) loga a 6
b) ln e пЂ­8
Simplify.
пЃ· A)
пЃ· B)
7
e
log 7 w
ln 8
Change of Base Formula
пЃ· 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.
пЃ· For ease of computing a logarithm, we may want to
switch from one base to another using the formula
log b M пЂЅ
log M
log b
or log b M пЂЅ
ln M
ln b
See Examples 7 & 8, page 406-7.
пЃ· Check Point 7: Use common logs to evaluate
log7 2506.
пЃ· Check Point 8: Use natural logs to evaluate
log7 2506.
Summary of
Properties of Logarithms
For a > 0, a п‚№ 1,and any real num ber k,
1) lo g
a
a пЂЅ 1,
2 ) lo g a 1 = 0 ,
ln e = 1
ln 1 = 0
A d d it 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
4) a
lo g a k
k
= k
= k, k > 0
Summary of
Properties of Logarithms (cont.)
F or x > 0 , y пЂѕ 0, a пЂѕ 0, a п‚№ 1 ,a n d a n y re a l n um b e r r,
5) Pr od uct R ule
6) Q uotie n t R ule
7 ) Pow e r R ule
log a x y = log a x + log a y
log a
x
y
пЂЅ log a x - log a y
r
log a x пЂЅ r log a x
```
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