Download 1 log log log MN M N = + log 9 2 log 9 log 2 ∙ = + log 100000 log

Section 3.4: Properties of Logarithms
§1 Some Basic Properties
=
log a 1 0=
and log a a 1
=
a log z M M
=
and log a a r r
§2 Product Rule
Let b, M, and N be positive real numbers with
b ≠ 1 . Then log b =
( MN ) logb M + logb N . The
product rule for logarithms states that the logarithm of a product is the sum of the logarithms.
For example,
log 3 ( 9 ⋅ 2=
) log3 9 + log3 2 . Note that we can simplify the first log expression. The
2 + log 3 2 .
answer is
For another example, note that
log10 100000 = 5 . We can expand this expression to get
log=
log10 (1000 ⋅ 100 ) . Use the product rule to get log10 1000 + log10 100 , which
10 100000
equals 3 + 2 = 5.
§3 Quotient Rule
Let b, M, and N be positive real numbers with
M 
b ≠ 1 . Then log b=
  log b M − log b N . The
N
quotient rule for logarithms states that the logarithm of a quotient is the difference of logarithms.
 e2 
For example, ln 
ln e 2 − ln 5 =−
2 ln 5 .
=
5
 
§4 Power Rule
Let b, M, and N be positive real numbers with
b ≠ 1 , and let p be any real number.
Then
log b M = p log b M .
p
For example,
Also note,
log 7 2 = 2log 7 . We use this rule to ‘pull the exponent to the front.’
ln=
x ln=
x1 2
1
ln x
2
We use all three of these rules to expand logarithmic expressions. Usually, when there are 2 or more
variables in a single log, we expand the expression as much as possible. It’s always best to convert any
expressions in radical form to exponential form first. Also, we usually use the power rule last.
3
For example, expand the following expression:
log 4
x
16 y 2
x1 3
Solution: First convert the radical to exponential form to get log 4
. Then use quotient rule to get
16 y 2
log 4 x1 3 − log 4 (16 y 2 ) . Then use product rule to get log 4 x1 3 − (log 4 16 + log 4 y 2 ) . Finally,
distribute the negative and use power rule to get
expression to get
1
log 4 x − log 4 16 − 2log 4 y . Simplify the log
3
1
log 4 x − 2 − 2log 4 y .
3
PRACTICE
 25 x3 
1) Expand as much as possible: log 5 
 3 y 


 32 x 3 
2) Expand as much as possible: log 2 
4 
 9 yz 
3) Expand as much as possible:
log 3
100 x
y2
§5 Condensing Logarithmic Expressions
Conversely, we can write the sum or difference of two or more logs as a single log expression. We use
the properties of logs. Make sure that you only condense logs whose bases are the same.
For example, we can condense
log b 12 + log b x =
log b 12 x .
Note, the coefficient of any log must be 1 before you can condense. This means using the power rule
first!
 x3 
1
3
12
For example, we can condense 3ln x − ln y =ln x − ln y =ln  1 2 
2
y 
PRACTICE
4) Condense into a single logarithm:
3log ( x + 4 ) − 2log 4 y
5) Condense into a single logarithm:
1
1
( log 6 x + log 6 y ) − 2log 6
2
36
6) Condense into a single logarithm:
3
log x − log y − 2log( z + 1)
2
§6 Change of Base Property
For any logarithmic bases a and b, and any positive number M,
log b M =
log a M
log a b
. Note that a can be any real number. When using the change of base formula, we
usually let a = 10 or e, meaning we can use the change of base formula to convert either the common
log or the natural logarithm.
For example, we can evaluate log 3 15 =
log10 15
ln15
. It does not matter which one you use –
or
log10 3
ln 3
both answers are the same.
For example, we know that
log 4 64 = 3 . It turns out that if we do
ln 64
, the answer is 3!
ln 4
§7 Common Mistakes
Some common errors:
log b ( M + N ) ≠ log b M + log b N
log b ( M − N ) ≠ log b M − log b N
log b ( MN ) ≠ log b M ⋅ log b N
 M  log b M
log b   ≠
 N  log b N
log b M
≠ log b M − log b N
log b N
log b ( MN ) ≠ p log b ( MN )
p
Do not make these mistakes! Make sure you review the product rule, quotient rule and power rule
carefully!