Given:
y   log
a
x
…and given that the point (2, -6) is on the graph
of the above log function, solve for a.
4.4 Properties of Logarithms
Properties of Logarithms:
Let b, m and n be positive numbers, b does not equal 1:
Product Property
log b mn  log b m  log b n
m
Quotient Property log b  log b m  log b n
n
Power Property
log b m  n log b m
n
Expand
5x
log 6
y
3

ln x 1  x 2

EXAMPLE 3
Condense
log9 + 3log2 – log3
1
 1
log 2    log 2  2 
 x
x 
Change of Base Formula:
Derivation:
How can we
solve for y?
log 2 7  y
Change of Base Formula:
If a, b and c are positive numbers with b and c not equal to one:
log a
log c a 
log c
And
ln a
log c a 
ln c
EXAMPLE
4
Evaluate
log 3 8
 1.893
log
8
5
 2.584
20
log 5
6
 .7481
Suppose ln 4 = c and ln 5 = d. Use properties of logs to write
each logarithm in terms of c and d.
ln 20
cd
log 4 25
2d
c
Log Properties’ Game
•Each table group is a team.
•First team to give the correct answer wins the
round.
•Team must articulate the answer using proper
math language.
Expand Completely:
 3 x2 1 

log 5  2
 x 1 


Condense Completely:
2 log 3 u  log 3 v
Condense Completely:
 x  2x  3 
 x  7x  6 



log

log
2

 x2 
x

4




2
2
Expand Completely:
 x  4 
ln  2

 x 1 
2
2
3
Expand Completely:
 5x 2 3 1  x 
ln 
2 
 4x  1 
Condense Completely:
21 log 3 x  log 3 9 x  log 5 25
3
2
Condense Completely:




1
1
3
2
log x  1  log x  1
3
2