Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of...

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Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Transcript of Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of...

Page 1: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Lesson 3.3, page 400Properties of Logarithms

Objective: To learn and apply the properties of logarithms.

Page 2: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Real-World Connection

Logarithms are used in applications involving sound intensity &

decibel level.

Page 3: Lesson 3.3, page 400 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?

Page 4: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

PRODUCT RULE, page 400

Product Property: logb(MN) = logbM + logbN

The logarithm of a product is the sum of the logarithms of the factors.

Ex) logbx3 + logby =

Page 5: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

See Example 1, pg. 401

Express as a single logarithm:2

3 3log logx w

Page 6: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Check Point 1

Use the product rule to expand each logarithmic expression:

A) log6(7 11) B) log(100x)

Page 7: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

QUOTIENT RULE, page 401

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 =

Page 8: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

See Example 2, page 402.

Express as a difference of logarithms. 10

loga b

Page 9: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Check Point 2

Use the quotient rule to expand each logarithmic expression:

5

8

23A) log B) ln

11

e

x

Page 10: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

POWER RULE, page 402

Power Property: logbMp = p logbM

The logarithm of a power of M is the exponent times the logarithm of M.

Ex) log2x3 =

Page 11: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

See Example 3, page 403.

Express as a product.

3log 7a

Page 12: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Check Point 3

Use the power rule to expand each logarithmic expression:

9 236A) log 3 B) ln C) log( 4)x x

Page 13: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Extra Practice

Express as a product.5log 11a

1/ 55log 11 log 11

1log 11

5

a a

a

Page 14: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Expanding Logarithmic Expressions(See blue box on page 403.)

Use properties of logarithms to change one logarithm into a sum or difference of others.

Example

)(log4)(log4

1)2(log2

)(log4)(log4

1)2(log)6(log

)(log4)(log4

1)236(log

)(loglog72log72

log

666

6662

6

666

46

4

1

664

4

6

yx

yx

yx

yxy

x

Page 15: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

See Example 4, page 404

Check Point 4: Use log properties to expand each expression as much as possible.

4 35 3

a) log ( ) b) log25b

xx y

y

Page 16: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Expanding Logs – Express as a sum or difference.

3 4

2loga

w y

z

Page 17: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

More Practice Expanding

a) log27b

b) log(y/3)2

c) log7a3b4

Page 18: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Condensing Logarithmic Expressions(See blue box on page 404.)

We can also use the properties of logarithms to condense expressions or “write as a single logarithm”.

See Example 5, page 404.

Page 19: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Let’s reverse things.

Express as a single logarithm.

log 125 log 25w w

Page 20: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Pencils down. Watch and listen.

Express as a single logarithm.

Solution:

16log 2log log

3b b bx y z

6 2 1/ 3

61/ 3

2

6 1/ 3 6 3

2 2

16log 2log log log log log

3

log log

log , or log

b b b b b b

b b

b b

x y z x y z

xz

y

x z x z

y y

Page 21: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Check Point 5

Write as a single logarithm.

a) log 25 log 4 b) log(7 6) log x x

Page 22: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Check Point 6Write as a single logarithm.

1a) 2 ln ln( 5) b) 2 log( 3) log

3x x x x

Page 23: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Check Point 6Write as a single logarithm.

1) log 2log 5 10log

4 b b bc x y

Page 24: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

More Practice

d) Write 3log2 + log 4 – log 16 as a single logarithm.

e) Can you write 3log29 – log69 as a single logarithm?

Page 25: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Review of Properties(from Lesson 3.2)

The Logarithm of a Base to a PowerFor 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 PowerFor any base a and any positive real number x,

(The number a raised to the power loga x is x.)

log .a xa x

Page 26: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Examples

Simplify.a) loga a 6

b) ln e 8

Page 27: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Simplify.

A)

B)

7log7 w

ln8e

Page 28: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

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 lnlog or log

log lnb b

M MM M

b b

Page 29: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

See Examples 7 & 8, page 406-7.

Check Point 7: Use common logs to evaluate log7 2506.

Check Point 8: Use natural logs to evaluate log7 2506.

Page 30: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Summary of Properties of Logarithms

a

a

ka

log k

For a>0, a 1,andany real number k,

ln e=1

2) log 1=0, ln1=0

Additional Logarithmic Properties

3) loga =k

4) a =k, k>0

1) log a 1,a

Page 31: Lesson 3.3, page 400 Properties of Logarithms Objective: To learn and apply the properties of logarithms.

Summary of Properties of Logarithms (cont.)

a a a

a a a

ra a

For x>0, y a a 1,and any real number r,

oduct Rule log xy= log x+log y

xQuotient Rule log log x- log y

y

7) Power Rule log x rlog x

0, 0,

5) Pr

6)