Properties of Logarithms

These properties are based on rules of exponents

since logs = exponents

I. Because in exponential form

(any number to the zero power = 1)

Example: = 5 to what power = 1?0

Example: = 0

II. Because in exponential form (any number to the first power is itself)

Example: = 5 to what power = 5?1

Example: = 1

III. Product Rule

Examples: = 𝑙𝑜𝑔𝑏 𝑥+ 𝑙𝑜𝑔𝑏 𝑦

=

𝑙𝑜𝑔𝑏𝑚𝑛=𝑙𝑜𝑔𝑏𝑚+ 𝑙𝑜𝑔𝑏𝑛

𝑙𝑜𝑔2+𝑙𝑜𝑔36 =𝑙𝑜𝑔3 9+ 𝑙𝑜𝑔3𝑏

Because in exponential form

IV. Quotient Rule

Examples: = 𝑙𝑜𝑔5 𝑥−𝑙𝑜𝑔5 𝑦

=

𝑙𝑜𝑔𝑏𝑚𝑛 =𝑙𝑜𝑔𝑏𝑚−𝑙𝑜𝑔𝑏𝑛

𝑙𝑜𝑔2𝑎− 𝑙𝑜𝑔2 3 =𝒍𝒐𝒈𝟑𝟔+𝒍𝒐𝒈𝟑𝒃−𝒍𝒐𝒈𝟑𝟕

Because in exponential form

V. Power Rule

Examples: = 3 𝑙𝑜𝑔5 𝑥

𝑙𝑜𝑔𝑏𝑚𝑛=𝑛𝑙𝑜𝑔𝑏𝑚

=3 𝑙𝑜𝑔2𝑎+4 𝑙𝑜𝑔2𝑏

Because in exponential form

VI. Change of Base Formula

Example: = 𝑙𝑜𝑔9𝑙𝑜𝑔5

𝑙𝑜𝑔𝑏𝑚=𝑙𝑜𝑔𝑚𝑙𝑜𝑔𝑏

These properties remain the same when working with the natural log.

True or False:________1) 3log2log)32log( ______ 2) )26log(2log6log

________ 3) )4log(4log5 5

______ 4) 5log3log

53log

________ 5) )3log2(log4)32log( 4 _______6) )65log()6log()5log(

________ 7) 5log3log5log3log

______ 8) )65log(6log5log

________ 9) 2log8log8log2

______ 10) 3)2(log 32

_______ 11) 3log42log)32log( 4 ______ 12) 2log2ln e

True

True

True

True

True

True

True

False

False

False

False

False

Use properties of logarithms to determine if each of the following is true or false. Check your answers using your calculator

Use the properties of logs to expand the following expressions:

)5(log 310 yx1.

yx 103

1010 loglog5log

yx 101010 loglog35log

1. Apply Product Rule:

2. Apply Power Rule:

Use the properties of logs to expand the following expressions:

2.5

222 loglog4log yx

yx 222 log5log4log

1. Apply Product Rule:

2. Apply Power Rule:

)4(log 52 xy

Use the properties of logs to expand the following expressions:

3.

zxy 1010 loglog

zyx 101010 logloglog 2. Apply Product Rule:

zxy

10log 1. Apply Quotient Rule:

Use the properties of logs to expand the following expressions:

4.

21

5log ba

21

55 loglog ab2. Apply Product Rule:

1. Change radical to exponential form:

ab5log

3. Apply Power Rule:ab 55 log21log

Use the properties of logs to expand the following expressions:

5.

52 lnln yx

yx ln5ln2

2. Apply Product Rule:

52ln yx

3. Apply Power Rule:

Write as a single logarithmic expression.

5.

310

21

10 )1(loglog xx

3

21

10 )1(log

xx

1log3log21

1010 xx1. Apply Reverse Power Rule:

2. Apply Reverse Quotient Rule:

3. Change to radical form310 )1(

logxx

Write as a single logarithmic expression.

6.

)2)(2(log5 xx

2log)2(log 55 xx

2. Simplify

1. Apply Reverse Product Rule:

)4(log 25 x

Write as a single logarithmic expression.

6.

35 lnln yx

)ln(3)ln(5 yx

2. Apply Reverse Product Rule:

)ln( 35 yx

1. Apply Reverse Power Rule:

Practice Time