Download log x y z − log ( ) log log mn m n

Math 152: Peacemaker
Blitzer/9.4
Properties of Logarithms
Let b ,
m , and n be positive real numbers with b ≠ 1
Rule
1.
log b (mn) = logb m + log b n
2.
m
log b   = log b m − log b n
n
3.
log b m p = p log b m
Example
1. Using Logarithmic Properties to Expand Logarithmic Expressions
Example 1
Use the properties of logarithms to expand each expression and evaluate if possible.
a.
log 5 25x
c.
log 9
e.
ln x 2 y 3
g.
4
log8 ( x − y ) z 6 


x
9
b.
ln(4 x)
d.
log 3 ( x + 3)
f.
3x
log  5 
 y 
2
2. Using Logarithmic Properties to Condense Logarithmic Expressions
Example 2
Use the properties of logarithms to condense each expression.
a.
log 3 486 − log3 18
b.
log 4 8 + log 4 32
c.
ln x − 4 ln y
d.
1
8 log b y + log b z
4
e.
1
3log 5 x + log 5 y − 4 log 5 z
2
3. Change of Base Property
For any logarithmic bases
a and b , and a real number M > 0 ,
log b M =
log a M
, so for practical purposes we
log a b
use the common logarithm or the natural logarithm:
log b M =
Example 3
a.
log8 35
b.
log16 5
log M
log b
log b M =
ln M
ln b
Use the change of base formula to evaluate the logarithms.
Math 152 – Blitzer/9.4
2