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Properties of Logarithms - Numeric
Contributed by: Potter
Remember that logbu (or similarly logbv) is another way to writean exponent.logbu is that exponent you need to raise base "b" to in order t get "u".
So the log properties below are just the exponent properties from before.
logb
Let b, u, and v be positive numbers such that b ≠ 1.
bases
multiplied
together
Product Property
u
v
= logb
add
exponents
u
+ logb
v
Properties of logarithms
logb
Quotient Porperty
bases
divided
u
v
= logb
u
subtract
exponents
- logb
v
a power
to a
power
logb
Power Property
u
n
=
n
multiply
exponents
logb
u
log556=log57•8
log556=log57+log58log556≈1.2 + 1.3≈2.5
Use the properties of logarithms and the values below to estimate thevalue of the logarithm below. Do not use a calculator to evaluate the log.
Product Property
Properties of logarithms
log7log7log7
Quotient Porperty
12
10
12
10
12
10
=log712-log710
≈1.3 - 1.2
≈0.1
log364=log382
log364=2•log38
log364≈2•1.9≈3.8
Power Property
Use the properties of logarithms and the values below to estimate thevalue of the logarithm below. Do not use a calculator to evaluate the log.
Answer:
Properties of logarithms
Round answer to one decimal.
If your answer looks like:
2.1    enter    2.15       enter    5.0.3      enter    0.3-4.2  enter   -4.2
Use the properties of logarithms and the values below to estimate thevalue of the logarithm below. Do not use a calculator to evaluate the log.
Answer:
Properties of logarithms
Round answer to one decimal.
If your answer looks like:
2.1    enter    2.15       enter    5.0.3      enter    0.3-4.2  enter   -4.2
Use the properties of logarithms and the values below to estimate thevalue of the logarithm below. Do not use a calculator to evaluate the log.
Answer:
Properties of logarithms
Round answer to one decimal.
If your answer looks like:
2.1    enter    2.15       enter    5.0.3      enter    0.3-4.2  enter   -4.2
Use the properties of logarithms and the values below to estimate thevalue of the logarithm below. Do not use a calculator to evaluate the log.
Answer:
Properties of logarithms
Round answer to one decimal.
If your answer looks like:
2.1    enter    2.15       enter    5.0.3      enter    0.3-4.2  enter   -4.2
Use the properties of logarithms and the values below to estimate thevalue of the logarithm below. Do not use a calculator to evaluate the log.
remember: log51 = 0
Answer:
Properties of logarithms
Round answer to one decimal.
If your answer looks like:
2.1    enter    2.15       enter    5.0.3      enter    0.3-4.2  enter   -4.2
Use the properties of logarithms and the values below to estimate thevalue of the logarithm below. Do not use a calculator to evaluate the log.
Answer:
Properties of logarithms
Round answer to one decimal.
If your answer looks like:
2.1    enter    2.15       enter    5.0.3      enter    0.3-4.2  enter   -4.2
Use the properties of logarithms and the values below to estimate thevalue of the logarithm below. Do not use a calculator to evaluate the log.
Answer:
Properties of logarithms
Round answer to one decimal.
If your answer looks like:
2.1    enter    2.15       enter    5.0.3      enter    0.3-4.2  enter   -4.2
Use the properties of logarithms and the values below to estimate thevalue of the logarithm below. Do not use a calculator to evaluate the log.
Answer:
Properties of logarithms
Round answer to one decimal.
If your answer looks like:
2.1    enter    2.15       enter    5.0.3      enter    0.3-4.2  enter   -4.2
remember: log51 = 0
Use the properties of logarithms and the values below to estimate thevalue of the logarithm below. Do not use a calculator to evaluate the log.
Answer:
Properties of logarithms
Round answer to one decimal.
If your answer looks like:
2.1    enter    2.15       enter    5.0.3      enter    0.3-4.2  enter   -4.2
Use the properties of logarithms and the values below to estimate thevalue of the logarithm below. Do not use a calculator to evaluate the log.
remember: log51 = 0
Answer:
Properties of logarithms
Round answer to one decimal.
If your answer looks like:
2.1    enter    2.15       enter    5.0.3      enter    0.3-4.2  enter   -4.2
=log72+log73+6log75
Often there are a combination of properties in one question.
Use the properties of logarithms to expand each expression.
There should be no powers or radicals in your answer.
Product Property
product & power
Properties of logarithms
=log723-log7318
=3•log72-18•log73
Quotient Porperty
quotient & power
=log5z2+log5x½=2•log5z+½•log5x=2•log5z+
power and product
Power Property
log5x
2
Use the properties of logarithms and the values below to expand eachexpression. There should be no powers or radicals in your answer.
Answer:
log4
Properties of logarithms
+
log4
answers should be in orderof increasing logs.i.e log67+log611+log612
not log612+log611+log67not log611+log67+log612
Use the properties of logarithms and the values below to expand eachexpression. There should be no powers or radicals in your answer.
Answer:
log8
Properties of logarithms
+
log8
answers should be in orderof increasing logs.i.e log67+log611+log612
not log612+log611+log67not log611+log67+log612
Use the properties of logarithms and the values below to expand eachexpression. There should be no powers or radicals in your answer.
Answer:
log6 
+log6 
Properties of logarithms
+log6 
answers should be in orderof increasing logs.i.e log67+log611+log612
not log612+log611+log67not log611+log67+log612
Use the properties of logarithms and the values below to expand eachexpression. There should be no powers or radicals in your answer.
Answer:
log2
Properties of logarithms
-
log2
Use the properties of logarithms and the values below to expand eachexpression. There should be no powers or radicals in your answer.
Answer:
log9
Properties of logarithms
-
log9
Use the properties of logarithms and the values below to expand eachexpression. There should be no powers or radicals in your answer.
Answer:
log3
Properties of logarithms
-
log3
Use the properties of logarithms and the values below to expand eachexpression. There should be no powers or radicals in your answer.
Remember that a radical is a fractional exponent. Rewrite as . . .
log3[5•(6•7)]log3[5]+log3(6•7)log3[5]+⅓log3(6•7)
Then write as a top:bottom fraction.
Answer:
log3
+log3
Properties of logarithms
+log3
answers should be in orderof increasing logs.i.e log67+log611+log612
not log612+log611+log67not log611+log67+log612
Use the properties of logarithms and the values below to expand eachexpression. There should be no powers or radicals in your answer.
Answer:
log6
Properties of logarithms
-
log6
Use the properties of logarithms and the values below to expand eachexpression. There should be no powers or radicals in your answer.
Answer:
log6
Properties of logarithms
+
log6
answers should be in orderof increasing logs.i.e log67+log611+log612
not log612+log611+log67not log611+log67+log612
Use the properties of logarithms and the values below to expand eachexpression. There should be no powers or radicals in your answer.
Answer:
log7
Properties of logarithms
+
log7
Often there are a combination of properties in one question.
Use the properties of logarithms to condense each expression into asingle logarithm. There should be no + or - in your answer.
Product Property
=log6(3•2•7)=log6(3•=log6(3•
product & power
3
3
2•7
14
)
)
Properties of logarithms
Quotient Porperty
quotient & power
=log724-log7324=log724
324
power and product
Power Property
=log73+⅓log8=log73+log8=log(73•8)=log(73
3
8
)
Use the properties of logarithms to condense each expression into asingle logarithm. There should be no + or - in your answer.
Answer:
log3(
Properties of logarithms
)
Use the properties of logarithms to condense each expression into asingle logarithm. There should be no + or - in your answer.
Answer:
log8(
Properties of logarithms
)
Use the properties of logarithms to condense each expression into asingle logarithm. There should be no + or - in your answer.
Answer:
log4(
Properties of logarithms
)
Use the properties of logarithms to condense each expression into asingle logarithm. There should be no + or - in your answer.
Answer:
log6(
Properties of logarithms
)
Use the properties of logarithms to condense each expression into asingle logarithm. There should be no + or - in your answer.
Answer:
log8(
Properties of logarithms
)
Use the properties of logarithms to condense each expression into asingle logarithm. There should be no + or - in your answer.
Answer:
log7(
Properties of logarithms
)
Use the properties of logarithms to condense each expression into asingle logarithm. There should be no + or - in your answer.
Answer:
log8(
Properties of logarithms
)
Use the properties of logarithms to condense each expression into asingle logarithm. There should be no + or - in your answer.
Answer:
log9(
Properties of logarithms
)
Use the properties of logarithms to condense each expression into asingle logarithm. There should be no + or - in your answer.
Answer:
log3(
Properties of logarithms
)
Use the properties of logarithms to condense each expression into asingle logarithm. There should be no + or - in your answer.
Answer:
log2(
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: 
Your calculator only calculates logarithms with two bases:       ten ( base "10") and natural log, ln (base "e"). For other bases, use       the conversion below.
Properties of logarithms
logb(a) = log(a)
log(b)
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
= log15 = 1.176    log4      0.602
=1.953
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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