Download 9.4 Properties Logarithms

Math 152 — Rodriguez
Blitzer — 9.4
Properties of Logarithms
A. These properties will be used to solve logarithmic equations in section 9.5.
B. The Product Rule:
logb (MN)=logbM + logbN
b, M and N are positive numbers, b ≠ 1
Proof: Will be done on board.
Showing equality of property:
log 4 ( 4 !16 ) = log 4 4 + log 4 16
!expanding
!!!
"
log 4 ( 4 #16 ) = log 4 4 + log 4 16
log 4 64 = log 4 4 + log 4 16
=
condensing
$!!!
!
Example: Use the properties of logarithms to expand this logarithmic expression as much
as possible. Where possible, evaluate logarithmic expressions without using a
calculator.
log 3 ( 9x )
Example: Use the properties of logarithms to condense the logarithmic expressions as
much as possible. Write the expression as a single logarithm whose coefficient
is 1. Where possible, evaluate logarithmic expressions.
log 7 5 + log 7 x
!M$
& = logb M ' logb N
N%
C. The Quotient Rule: log b #"
b, M and N are positive numbers, b ≠ 1
! 27 $
& = log 3 27 ' log 3 3
3%
Showing equality of property: log 3 #"
Examples:
! x $
&
125 %
Expand: log 5 #"
Condense:
log 3 405 ! log 3 5
D. The Power Rule:
logb M p = p logb M
b, M and N are positive numbers, b ≠ 1;
p any real number
Showing equality of property: log 3 3 = 4 log 3 3
4
Expand: log 5 x
2
E. Expanding and Condensing Logarithmic Expressions
Example: Use the properties of logarithms to expand the logarithmic expressions as much
as possible. Where possible, evaluate logarithmic expressions without using a
calculator.
( )
1) log 5 xy 3
! x$
&
" 27 %
2) log 3 #
! 6$
&
e5 %
3) ln #"
Example: Use the properties of logarithms to condense the logarithmic expressions as much
as possible. Write the expression as a single logarithm whose coefficient is 1.
Where possible, evaluate logarithmic expressions.
4) 2 log x + 3log y
Blitzer — 9.4
5)
log 2 ( x + 5 ) ! log 2 ( x )
6) 5 log 2 x ! log 2 y
Page 2 of 2