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Properties of Logarithms
These properties are based on
rules of exponents
since
logs = exponents
I.
Because in exponential form
(any number to the zero power = 1)
Example: = 5 to what power = 1?0
Example: = 0
II.
Because in exponential form
(any number to the first power is itself)
Example: = 5 to what power = 5?1
Example: = 1
III. Product Rule
Examples: = 𝑙𝑜𝑔𝑏 𝑥+ 𝑙𝑜𝑔𝑏 𝑦
=
𝑙𝑜𝑔𝑏𝑚𝑛=𝑙𝑜𝑔𝑏𝑚+ 𝑙𝑜𝑔𝑏𝑛
𝑙𝑜𝑔2+𝑙𝑜𝑔36 =
𝑙𝑜𝑔3 9+ 𝑙𝑜𝑔3𝑏
Because in exponential form
IV. Quotient Rule
Examples: = 𝑙𝑜𝑔5 𝑥−𝑙𝑜𝑔5 𝑦
=
𝑙𝑜𝑔𝑏𝑚𝑛
=𝑙𝑜𝑔𝑏𝑚−𝑙𝑜𝑔𝑏𝑛
𝑙𝑜𝑔2𝑎− 𝑙𝑜𝑔2 3 =
𝒍𝒐𝒈𝟑𝟔+𝒍𝒐𝒈𝟑𝒃−𝒍𝒐𝒈𝟑𝟕
Because in exponential form
V. Power Rule
Examples: = 3 𝑙𝑜𝑔5 𝑥
𝑙𝑜𝑔𝑏𝑚𝑛=𝑛𝑙𝑜𝑔𝑏𝑚
=
3 𝑙𝑜𝑔2𝑎+4 𝑙𝑜𝑔2𝑏
Because in exponential form
VI. Change of Base Formula
Example: =𝑙𝑜𝑔9𝑙𝑜𝑔5
𝑙𝑜𝑔𝑏𝑚=𝑙𝑜𝑔𝑚𝑙𝑜𝑔𝑏
These properties remain the same when working with the natural log.
True or False:
________1) 3log2log)32log( ______ 2) )26log(2log6log
________ 3) )4log(4log5 5
______ 4) 5log3log
5
3log
________ 5) )3log2(log4)32log( 4 _______6) )65log()6log()5log(
________ 7) 5log3log5log
3log
______ 8) )65log(6log5log
________ 9) 2log
8log8log2
______ 10) 3)2(log 32
_______ 11) 3log42log)32log( 4 ______ 12) 2log2ln e
True
True
True
True
True
True
True
False
False
False
False
False
Use properties of logarithms to determine if each of the following is true or false. Check your answers using your calculator
Use the properties of logs to expand the following expressions:
)5(log 310 yx1.
yx 103
1010 loglog5log
yx 101010 loglog35log
1. Apply Product Rule:
2. Apply Power Rule:
Use the properties of logs to expand the following expressions:
2.
5222 loglog4log yx
yx 222 log5log4log
1. Apply Product Rule:
2. Apply Power Rule:
)4(log 52 xy
Use the properties of logs to expand the following expressions:
3.
zxy 1010 loglog
zyx 101010 logloglog 2. Apply Product Rule:
z
xy10log
1. Apply Quotient Rule:
Use the properties of logs to expand the following expressions:
4.
2
1
5log ba
2
1
55 loglog ab2. Apply Product Rule:
1. Change radical to exponential form:
ab5log
3. Apply Power Rule:ab 55 log2
1log
Use the properties of logs to expand the following expressions:
5.
52 lnln yx
yx ln5ln2
2. Apply Product Rule:
52ln yx
3. Apply Power Rule:
Write as a single logarithmic expression.
5.
310
2
1
10 )1(loglog xx
3
2
1
10 )1(log
xx
1log3log2
11010 xx
1. Apply Reverse Power Rule:
2. Apply Reverse Quotient Rule:
3. Change to radical form310 )1(
logxx
Write as a single logarithmic expression.
6.
)2)(2(log5 xx
2log)2(log 55 xx
2. Simplify
1. Apply Reverse Product Rule:
)4(log 25 x
Write as a single logarithmic expression.
6.
35 lnln yx
)ln(3)ln(5 yx
2. Apply Reverse Product Rule:
)ln( 35 yx
1. Apply Reverse Power Rule:
Practice Time
