3.3 Properties of Logarithms

Students will rewrite logarithms with different bases.

Students will use properties of logarithms to evaluate or rewrite logarithmic expressions.

Students will use properties of logarithms to expand or condense logarithmic expressions.

Students will use logarithmic functions to model and solve real-life problems.

Properties of Logarithms1. If then

2.

3.

4.

5.

loga 1 0

log ( ) log logb b buv u v

log log logb b b

u

vu v

log logbx

bu x u

y xblog b xy

Example

Solve

Solve

log4 567

logb a

Change of Base Formula

loglog

log

ln

ln

log

logab

b

xx

a

x

a

x

a

log4 567

Example 1

Change base using common log.

a. b. log4 25 log2 12

Example 2

Change base using natural log.

a. b. log4 25 log2 12

Example of log ( ) log logb b buv u v

log (( )( )) log log10 10 10100 1000 100 1000

Example of

log ( ) log log10 10 10

100

1000100 1000

log log logb b b

u

vu v

Example 3

Write each logarithm in terms of 2 and 3.

a. b. ln2

27ln 6

Example 4

Use the properties of logarithms to verify the statement:

log log10 10

1

100100

Example 5

Use the properties of logarithms to expand each expression:

a. b. log4

35x yln

3 5

7

x

Example 6Use the properties of logarithms to condense each logarithmic expression:

a. b. c.1

23 110 10log log ( )x x

2 2ln( ) lnx x 1

342 2log log ( )x x