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Properties of Logarithms
Contributed by: LI
(Original author: Potter)
logb
Let b, u, and v be positive numbers such that b ≠ 1.

Here, b can be any positive numbers other than 10.
u and vmultipliedtogether
Product Property
u
v
乘變加
= logb
add
exponents
u
+ logb
v
Properties of logarithms
logb
Quotient Porperty
u and vdivided
u
v
= logb
除變減
u
subtract
exponents
- logb
v
u to apower
logb
Power Property
連乘變連加
u
n
=
n
multiply
exponents
logb
u
Estimate the following logarithm WITHOUT using a calculator.
log556 = log5(7•8)= log57 + log58≈ 1.2 + 1.3≈ 2.5
Product Property
Properties of logarithms
= log712 - log710≈ 1.3 - 1.2≈ 0.1
Quotient Porperty
log7
12
10
log364 = log382
            = 2•log38
            ≈ 2•1.9             ≈ 3.8
Power Property
Estimate the following logarithm WITHOUT using a calculator.
Answer:
Properties of logarithms
Hint: Go back to the previous page.
Estimate the following logarithm WITHOUT using a calculator.
Answer:
Properties of logarithms
Estimate the following logarithm WITHOUT using a calculator.
Answer:
Properties of logarithms
Estimate the following logarithm WITHOUT using a calculator.
Answer:
Properties of logarithms
Remember: log51 = 0
Estimate the following logarithm WITHOUT using a calculator.
Answer:
Properties of logarithms
Estimate the following logarithm WITHOUT using a calculator.
Answer:
Properties of logarithms
Estimate the following logarithm WITHOUT using a calculator.
Answer:
Properties of logarithms
Estimate the following logarithm WITHOUT using a calculator.
Answer:
Properties of logarithms
Remember: log91 = 0
Estimate the following logarithm WITHOUT using a calculator.
Answer:
Can you make 36 two different ways?
Do you get the same answer either way? :)
Properties of logarithms
Answer (yes/no):
Estimate the following logarithm WITHOUT using a calculator.
Answer:
Properties of logarithms
Often there are a combination of properties in one question.
Expand the followings. No powers or radicals in your answer.
= log9x+ log7y4
= 2•log9+ 4•log7y
product & power
Product Property
Properties of logarithms
= log7x16 - log7y4
=16•log7- 4•log7y
quotient & power
Quotient Porperty
= log5z+ log5x½= 2•log5+ ½•log5x
= 2•log5+
power and product
Power Property
log5x
2
Expand the followings. No powers or radicals in your answer.
Answer:
log8
Properties of logarithms
+
log8
Hint: Go back to the previous page.
answers should be in orderof increasing logs.i.e log6x+log6y+log6z
not log6z+log6y+log6xnot log6y+log6x+log6z
Expand the followings. No powers or radicals in your answer.
Answer:
log7
Properties of logarithms
+
log7
answers should be in orderof increasing logs.i.e log6x+log6y+log6z
not log6z+log6y+log6xnot log6y+log6x+log6z
Expand the followings. No powers or radicals in your answer.
Answer:
log6
+
log6
Properties of logarithms
+
log6
answers should be in orderof increasing logs.i.e log6x+log6y+log6z
not log6z+log6y+log6xnot log6y+log6x+log6z
Expand the followings. No powers or radicals in your answer.
Answer:
log2
Properties of logarithms
-
log2
Expand the followings. No powers or radicals in your answer.
Answer:
log9
Properties of logarithms
-
log9
Expand the followings. No powers or radicals in your answer.
Answer:
log9
Properties of logarithms
-
log9
Expand the followings. No powers or radicals in your answer.
Answer:
log
+
log
Properties of logarithms
+
log
Hint:
log(u•v•w)½ = ½log(u•v•w)
Then write as a fraction.
Expand the followings. No powers or radicals in your answer.
Answer:
log2
Properties of logarithms
+
log2
Expand the followings. No powers or radicals in your answer.
Answer:
log3
Properties of logarithms
+
log3
Expand the followings. No powers or radicals in your answer.
Answer:
log7
+
Properties of logarithms
log7
Often there are a combination of properties in one question.
Simplify the followings. No + or - in your answer.
Product Property
=log8(u•v•w)=log8(
product & power
3
u•v•w
)
Properties of logarithms
Quotient Porperty
quotient & power
=log7x3-log7y2=log7x3
y2
power and product
Power Property
=log5c3log5a=log5c3+log5a½=log5(c3•a½)=log5(c3
a
)
Simplify the followings. No + or - in your answer.
Answer:
log5(
Properties of logarithms
)
Simplify the followings. No + or - in your answer.
Answer:
log4(
Properties of logarithms
)
Simplify the followings. No + or - in your answer.
Answer:
log9(
Properties of logarithms
)
Simplify the followings. No + or - in your answer.
Answer:
log2(
Properties of logarithms
)
Simplify the followings. No + or - in your answer.
Answer:
log4(
Properties of logarithms
)
Simplify the followings. No + or - in your answer.
Answer:
log4(
Properties of logarithms
)
Simplify the followings. No + or - in your answer.
Answer:
log6(
Properties of logarithms
)
Simplify the followings. No + or - in your answer.
Answer:
log(
Properties of logarithms
)
Simplify the followings. No + or - in your answer.
Answer:
log4(
Properties of logarithms
)
Use the the change of base formula and your calculator to approximatethe value to the nearest thousandths.
i.e. answers should look like 3.456 or 0.123 or -6.789
Change of base formula: 
logb(a) = 
Properties of logarithms
log(b)
log(a)
Use the the change of base formula and your calculator to approximatethe value to the nearest thousandths.
i.e. answers should look like 3.456 or 0.123 or -6.789
Change of base formula:Example:
logb(a) = log(a)
log(b)
Properties of logarithms
=
=1.953
log15
log4
=
1.176
0.602
Use the the change of base formula and your calculator to approximatethe value to the nearest thousandths.
i.e. answers should look like 3.456 or 0.123 or -6.789 or 6.000
=
=
=
Properties of logarithms
=
=
Use the the change of base formula and your calculator to approximatethe value to the nearest thousandths.
i.e. answers should look like 3.456 or 0.123 or -6.789 or 6.000
=
=
Properties of logarithms
=
=
=
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