Post on 18-Jan-2016
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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