Section 3.3
Properties of Logarithms
The Product Rule
Example
Use the product rule to expand each logarithmic expression.
log3 (9  5)
log (1000x)
The Quotient Rule
Example
Use the quotient rule to expand each logarithmic expression.
 25 
log 5  
 x 
x
log  
8
 e3 
ln  
7
The Power Rule
When we use the power rule to "pull the exponent to the front"
we say that we are expanding a logarithmic expression. For example
we can use the power rule to expand ln x 2:
Example
Use the power rule to expand each logarithmic expression.
log 5 7 2
log 2 (8 x) 4
log x
ln(6e)5
Expanding Logarithmic
Expressions
Study Tip
Example
Use logarithmic properties to expand each expression as
much as possible.


 25 
log 5  
 y 
 10 
ln  
 e 
3
log b x z

log 9 10

2
Condensing Logarithmic
Expressions
Example
Write as a single logarithm (condense).
log 2 4  log 2 8
log  6 x   log 6
2 log 3 9  log 3 27
ln  x  2   5ln x
The Change-of-Base Property
Graphing Calculator
Example
Use common logarithms to evaluate log 412.
Use your calculator.
Use properties of logarithms to expand each expression.
Where possible, evaluate without a calculator.
 81 
log 9  
 x
(a) 2  x
(b) 2  log 9 x
(c)
2
log 9 x
2
(d)
x
Use properties of logarithms to expand each expression.
Where possible, evaluate without a calculator.
log 3 27 y
(a) 3log 3 y
(b) log 3  log 3 y
(c) log 3  log 3 y
(d) 1  log 3 y