Honors Algebra 2 1

Properties of Logarithms

During this lesson, you will:

Expand the logarithm of a product, quotient, or power

Simplify (condense) a sum or difference of logarithms

Honors Algebra 2 2

Part 1:

Expanding Logarithms

Mrs. McConaughy Honors Algebra 2 3

The Product Rule

Let M, N, and b be any positive numbers, such that b ≠ 1.

log b (M ∙ N ) = log b M+ log b N

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

Connection: When we multiply exponents with a common base, we add the exponents.

PROPERTY: The Product Rule (Property)

4

Example Expanding a Logarithmic Expression Using

Product Rule

log (4x) = log 4 + log xThe logarithm of a product

is

The sum of the logarithms.

Use the product rule to expand:

a.log4 ( 7 • 9) = _______________

b. log ( 10x) = ________________

= ________________

log4 ( 7) + log 4(9)

log ( 10) + log (x)

1 + log (x)

Honors Algebra 2 5

Property: The Quotient Rule (Property)

The Quotient RuleLet M, N, and b be any positive

numbers, such that b ≠ 1.

log b (M / N ) = log b M - log b N

The logarithm of a quotient is the difference of the logarithms.

Connection: When we divide exponents with a common base, we subtract the exponents.

Mrs. McConaughy 6

log (x/2) = log x - log 2

Example Expanding a Logarithmic Expression Using

Quotient Rule

The logarithm of a quotient

is

The difference of the logarithms.

Use the quotient rule to expand:

a.log7 ( 14 /x) = ______________

b. log ( 100/x) = ______________

= ______________

log7 ( 14) - log 7(x)

log ( 100) - log (x)

2 - log (x)

Honors Algebra 2 7

PROPERTY: The Power Rule (Property)

The Power Rule

Let M, N, and b be any positive numbers, such that b ≠ 1.

log b Mx = x log b M

When we use the power rule to “pull the exponent to the front,” we say we are _________ the logarithmic expression.expanding

Honors Algebra 2 8

Example Expanding a Logarithmic Expression Using Power Rule

Use the power rule to expand:

a.log5 74= _______________

b. log √x = ________________

= ________________

4log5 7

log x 1/2

1/2 log x

Honors Algebra 2 9

Summary: Properties for Expanding Logarithmic

Expressions

Properties of Logarithms

Let M, N, and b be any positive numbers, such that b ≠ 1.

Product Rule:

Quotient Rule:

Power Rule:

log b (M ∙ N ) = log b M+ log b N

log b (M / N ) = log b M - log b N

log b Mx = x log b M

NOTE: In all cases, M > 0 and N >0.

Honors Algebra 2 10

Check Point: Expanding Logarithmic Expressions

Use logarithmic properties to expand each expression:

a. logb x2√y b. log6 3√x

36y4

log b x2 + logb y1/2

2log b x + ½ logb y

log 6 x1/3 - log636y4

log 6 x1/3 - (log636 + log6y4)

1/3log 6 x - log636 - 4log6y

2

Honors Algebra 2 11

Check Point: Expanding Logs

Expand:

log 2 3xy2

log 8 26(xy)2

= log 2 3 + log 2 x + 2log 2 y

= log 8 26 + log 8 x2 + log 8 y2

= 6log 8 2 + 2log 8 x + 2log 8 y

NOTE: You are expanding, not condensing (simplifying) these logs.

Honors Algebra 2 12

Part 2: Condensing (Simplifying) Logarithms

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Part 2: Condensing (Simplifying) Logarithms

To condense a logarithm, we write the sum or difference of two or more logarithms as single expression.

NOTE: You will be using properties of logarithms to do

so.

Honors Algebra 2 14

Properties for Condensing Logarithmic Expressions (Working Backwards)

Properties of Logarithms

Let M, N, and b be any positive numbers, such that b ≠ 1.

Product Rule:

Quotient Rule:

Power Rule:

log b M+ log b N = log b (M ∙ N)

log b M - log b N = log b (M /N)

x log b M = log b Mx

Honors Algebra 2 15

Example Condensing Logarithmic Expressions

Write as a single logarithm: a. log4 2 + log 4 32 =

=a. log (4x - 3) – log x =

log 4 64 3

log (4x – 3)

x

Mrs. McConaughy Honors Algebra 2 16

NOTE: Coefficients of logarithms must be 1 before you condense them using the product and quotient rules.Write as a single

logarithm:

a. ½ log x + 4 log (x-1)

b. 3 log (x + 7) – log x

c. 2 log x + log (x + 1)

= log x ½ + log (x-1)4

= log √x (x-1)4

= log (x + 7)3 – log x= log (x + 7)3

x

= log x2 + log (x + 1)= log x2 (x + 1)

Honors Algebra 2 17

Check Point: Simplifying (Condensing) Logarithms

a.log 3 20 - log 3 4 =

b. 3 log 2 x + log 2 y =

c. 3log 2 + log 4 – log 16 =

log 3 (20/4) = log 3 5

log 2 x 3y

log 23 + log 4 – log 16 = log 32/16 =log 2

Honors Algebra 2 18

Example 1 Identifying the Properties of LogarithmsState the property or properties used to rewrite each expression:

Property:____________________________log 2 8 - log 2 4 = log 2 8/4 = log 2 2 = 1

Property:____________________________log b x3 y = log b x3 + log b 7 = 3log b x + log b 7

Property:____________________________

log 5 2 + log 5 6 = log 512

Quotient Rule (Property)

Product Rule/Power Rule

Product Rule (Property)

Sometimes, it is necessary to use more than one property of logs when you expand/condense an expression.

Honors Algebra 2 19

Example Demonstrating Properties of Logs

Use log 10 2 ≈ 0.031 and log 10 3 ≈ 0.477 to approximate the following:

a. log 10 2/3 b. log 10 6 c. log 10 9 log10 2 – log10 3

0.031 – 0.477

0.031 – 0.477

– 0.466

Change of Base Formula

• Example loglog558 =8 =

• This is also how you graph in another base. Enter y1=log(8)/log(5). Remember, you don’t have to enter the base when you’re in base 10!

log

log.

8

512900

b

logMlog M

logb

Examples

• Find the value of log2 37

• Change to base 10 and use your calculator. log 37/log 2

• Now use your calculator and round to hundredths.

= 5.21

• Log7 99 = ?

• Change to base 10.  Try it and see.

• log3 81

• log4 256

• log2 1024

Let’s try some• Working backwards now: write the following as a single

logarithm.

16log4log 44 nm 22 log4log2 2log5log

Let’s try something more complicated . . .

)xlogx(logxloglog 53525 4444

Condense the logsCondense the logslog 5 + log x – log 3 + 4log 5log 5 + log x – log 3 + 4log 5

Let’s try something more complicated . . .

2

4

y3

x10log

3

8 5

x2log

• Expand