12.3A Properties of Logarithms
Since logarithms are exponents, we should expect logarithms to have properties that correspond to the properties of exponents.
Product Rule
Quotient Rule
Power-of-a-Power Rule
Expanding Logarithms
The following three rules are used to expand logarithms.
mand n are any real number, variable or expression >0.
b>0 and b1.cis any real number.
Product Rule
Quotient Rule
Power Rule
NOTE: There is no property of logarithms that can be used to
simplify the log of a sum, or the log of a
difference,
To expand a logarithm, use the rules of logarithms to rewrite the logarithmic expression until all arguments are primeor simplified to a rational value.
Expand
The argument, 3x, is a product. Use the Product Rule.
log 3x = log 3 + log xall arguments are prime
Expand
The argument, 6,is not prime. Write it as a product of primes,
. Use the Product Rule.
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Expand
The argument,is a product. Although 10 is not prime, we do not factor it because the log is base–10. Use the Product Rule.
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Expand
The argument, 49,is not prime. Write it as a power, .
Use the Power Rule.
the argument, 7, is prime
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Expand
The argument,, is a quotient. Use the Quotient Rule.
all arguments are prime
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Expand
The argument, , is a power. Use the Power Rule.
the argument, x, is prime
If more than one rule must be applied, use the reverse order ofoperations to expand the logarithm.
Expand
The argument, ,contains a product and a power.
To simplify we would (1) do the power
(2) do the product
To expand, use the Product Rule then the Power Rule.
all arguments are prime
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Expand
The argument, ,contains a product and a power.
To simplify we would (1) do the product
(2) do the power
To expand, use the Power Rule then the Product Rule.
all arguments are prime
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Expand
The argument, ,contains a product and a power.
To simplify we would (1) do the product
(2) do the power
To expand, use the Power Rule then the Product Rule.
the argument, x, is prime
See Examples 1 – 4 on pages 845 – 848.
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Proofs of the Logarithm Properties
Product Rule
mis a power of a, say
nis a power of a, say
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Quotient Rule
mis a power of a, say
nis a power of a, say
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Power Rule
mis a power of a, say