Download 3.3_Properties of Logarithms

x x
2
3
x
2 3
x
5
Example 1
Use the product rule to expand each logarithmic expression.
ln  7 x 
log 4  7  5 
log3 (9  5)
log 10 x 
log (1000 x)
7
x
3
74

x

x
4
x
Example 2
Use the quotient rule to expand each logarithmic expression.
 x
log 5  
2
 25 
log5  
 x 
 19 
log 7  
 x
 x
log  
8
 e3 
ln  
 11 
 e3 
ln  
7
x 
3
5
x
53
x
15
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 3
Use the power rule to expand each logarithmic expression.
ln x
2
log 5 7 2
log5 7
4
log 2 (8 x) 4
log  4 x 
5
ln x
log x
ln(6e)5
Example 4

log b x
2
y

Example 5
 3x
log 6 
4
36
y




Do Now
Use logarithmic properties to expand each expression as much as possible.

log b x z

2
3
 25 
log 5    3log  25 
5

 y 
y
 
 3  log 5 25  log 5 y 
 3log 5 25  3log 5 y
 3  2   3log 5 y
 6  3log 5 y
 10 
ln    ln10  ln e
 e 
 ln10  1

log 9 10

 log 9  log 10
 log 9  log10
1
2
1
 log 9  log10
2
1
 log 9 
2
Page 449
# 1-35 odds
Check your answers in the
back of the text pg. AA37
Study Tip
Example 6
log 4 2  log 4 32
Example 7
log  4 x  3  log x
http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-625s.html
Extra
Write as a single logarithm (condense).
log  6 x   log 6
log 2 4  log 2 8
 log 2 4 8  log 2 8 2
1
2
 log 2 23  2  log 2 2
7
2
7
7
7
 log 2 2  (1) 
2
2
2
2 log 3 9  log 3 27
 log 3 92  log 3 27
92  log 81  log 3  1
 log3
3
3
27
27
6x
 log
6
 log x
ln  x  2   5ln x
 ln  x  2  ln x5
 ln  x  2   x 5  
 ln( x 6  2 x 5 )
Graphing 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