These properties are based on
rules of exponents
since
logs =
exponents
I.
𝑙𝑜𝑔𝑏 1 = 0
Because in exponential form
𝑏0 = 1
(any number to the zero power = 1)
Example:
Example:
0
𝑙𝑜𝑔𝑚 1= 0
𝑙𝑜𝑔5 1=
5 to what
power = 1?
II.
𝑙𝑜𝑔𝑏 𝑏 = 1
Because in exponential form
𝑏1 = 𝑏
(any number to the first power is itself)
Example:
Example:
1
𝑙𝑜𝑔𝑚 𝑚= 1
𝑙𝑜𝑔5 5=
5 to what
power = 5?
III. Product Rule
𝑙𝑜𝑔𝑏 𝑚𝑛 = 𝑙𝑜𝑔𝑏 𝑚 + 𝑙𝑜𝑔𝑏 𝑛
Because in exponential form 𝑏𝑚 × 𝑏𝑛 = 𝑏𝑚+𝑛
Examples:
𝑙𝑜𝑔𝑏 𝑥𝑦 = 𝑙𝑜𝑔𝑏 𝑥 + 𝑙𝑜𝑔𝑏 𝑦
𝑙𝑜𝑔6
= 𝑙𝑜𝑔2 + 𝑙𝑜𝑔3
𝑙𝑜𝑔3 9𝑏 = 𝑙𝑜𝑔3 9 + 𝑙𝑜𝑔3 𝑏
IV. Quotient Rule
𝑚
𝑙𝑜𝑔𝑏 = 𝑙𝑜𝑔𝑏 𝑚 − 𝑙𝑜𝑔𝑏 𝑛
𝑛
Because in exponential form
Examples:
𝑏𝑚
𝑏𝑛
= 𝑏𝑚−𝑛
𝑥
𝑙𝑜𝑔5 = 𝑙𝑜𝑔5 𝑥 − 𝑙𝑜𝑔5 𝑦
𝑦
𝑎
𝑙𝑜𝑔2 = 𝑙𝑜𝑔2 𝑎 − 𝑙𝑜𝑔2 3
3
6𝑏
𝑙𝑜𝑔3 = 𝒍𝒐𝒈𝟑 𝟔 + 𝒍𝒐𝒈𝟑 𝒃 − 𝒍𝒐𝒈𝟑 𝟕
7
V.
Power Rule
𝑛
𝑙𝑜𝑔𝑏 𝑚 = 𝑛𝑙𝑜𝑔𝑏 𝑚
Because in exponential form 𝑏𝑚
Examples:
𝑛
= 𝑏𝑚𝑛
= 3𝑙𝑜𝑔5 𝑥
𝑙𝑜𝑔5
3
𝑥
𝑙𝑜𝑔2
3
4
𝑎 𝑏 =
3𝑙𝑜𝑔2 𝑎 + 4𝑙𝑜𝑔2 𝑏
VI. Change of Base Formula
𝑙𝑜𝑔𝑚
𝑙𝑜𝑔𝑏 𝑚 =
𝑙𝑜𝑔𝑏
Example:
𝑙𝑜𝑔9
𝑙𝑜𝑔5 9 =
𝑙𝑜𝑔5
These properties remain the same when working with the natural log.
Use properties of logarithms to determine if each of the following is true or false.
Check your answers using your calculator
True
________1)
log( 2  3)  log 2  log 3
False 2)
______
log 6  log 2  log( 6  2)
True 3)
________
5 log 4  log( 45 )
True 4)
______
3
log    log 3  log 5
5
False 5)
________
False
log( 5)  log( 6)  log( 5  6)
log( 2  34 )  4(log 2  log 3) _______6)
False
________ 7)
log 3
 log 3  log 5
log 5
False 8)
______
True
________ 9)
log 8
log 2 8 
log 2
True 10)
______
log 2 (23 )  3
True
_______ 11)
log( 2  34 )  log 2  4 log 3
True
______ 12)
ln 2  log e 2
log 5  log 6  log( 5  6)
Use the properties of logs to expand the following expressions:
1.
log 10 (5 x 3 y )
log10 5  log10 x  log10 y
3
log10 5  3 log10 x  log10 y
1. Apply Product Rule:
2. Apply Power Rule:
Use the properties of logs to expand the following expressions:
2.
log 2 (4 xy5 )
log 2 4  log 2 x  log 2 y
5
log 2 4  log 2 x  5 log 2 y
1. Apply Product Rule:
2. Apply Power Rule:
Use the properties of logs to expand the following expressions:
3.
 xy 
log10  
 z 
1. Apply Quotient Rule:
log10 xy  log10 z
log10 x  log10 y  log10 z
2. Apply Product Rule:
Use the properties of logs to expand the following expressions:
4.
 
log 5 b a
log 5 ba
1
2
log 5 b  log 5 a
1. Change radical to exponential form:
1
2
1
log 5 b  log 5 a
2
2. Apply Product Rule:
3. Apply Power Rule:
Use the properties of logs to expand the following expressions:
5.

ln x 2 y 5

ln x 2  ln y 5
2. Apply Product Rule:
2 ln x  5 ln y
3. Apply Power Rule:
Write as a single logarithmic expression.
5.
1
log10 x  3 log10  x  1
2
1. Apply Reverse Power Rule:
1
2
log10 x  log10 ( x  1) 3
1
2
x
log10
( x  1) 3
x
log10
( x  1) 3
2. Apply Reverse Quotient Rule:
3. Change to radical form
Write as a single logarithmic expression.
6.
log 5 ( x  2)  log 5  x  2 
1. Apply Reverse Product Rule:
log 5 ( x  2)( x  2)
log 5 ( x 2  4)
2. Simplify
Write as a single logarithmic expression.
1. Apply Reverse Power Rule:
5 ln( x)  3 ln( y )
6.
ln x 5  ln y 3
ln( x 5 y 3 )
2. Apply Reverse Product Rule:
Practice Time