Lesson 3.3, page 400Properties of Logarithms

Objective: To learn and apply the properties of logarithms.

Real-World Connection

Logarithms are used in applications involving sound intensity &

decibel level.

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, 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 =

See Example 1, pg. 401

Express as a single logarithm:2

3 3log logx w

Check Point 1

Use the product rule to expand each logarithmic expression:

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

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 =

See Example 2, page 402.

Express as a difference of logarithms. 10

loga b

Check Point 2

Use the quotient rule to expand each logarithmic expression:

5

8

23A) log B) ln

11

e

x

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 =

See Example 3, page 403.

Express as a product.

3log 7a

Check Point 3

Use the power rule to expand each logarithmic expression:

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

Extra Practice

Express as a product.5log 11a

1/ 55log 11 log 11

1log 11

5

a a

a

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

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

Expanding Logs – Express as a sum or difference.

3 4

2loga

w y

z

More Practice Expanding

a) log27b

b) log(y/3)2

c) log7a3b4

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.

Let’s reverse things.

Express as a single logarithm.

log 125 log 25w w

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

Check Point 5

Write as a single logarithm.

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

Check Point 6Write as a single logarithm.

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

3x x x x

Check Point 6Write as a single logarithm.

1) log 2log 5 10log

4 b b bc x y

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 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

Examples

Simplify.a) loga a 6

b) ln e 8

Simplify.

A)

B)

7log7 w

ln8e

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

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.

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

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)