MAT106:
Trigonometry
Justifications/Proofs of Various Properties of Logarithms
Below are basic algebraic proofs of the validity of some of the basic properties of logarithms.
In order
to proceed, the following basic facts should be noted:
a.
is equivalent to
.
b. For any logarithmic expression
,
it is always true that
,
,
and
.
c.
For any positive values
and
,
there exists some value
such that
.
Power Rule
Consider the Power Rule:
.
Note that
for some value of
.
Rewriting the preceding logarithm in exponential form yields
.
Since
,
it can be stated that
for some value of
.
So,
can be rewritten as
(
)
.
Thus, by equating exponents (the technique for solving a Type I Exponential Equation),
.
But,
can be rewritten as
.
Hence,
is equivalent to
.
Therefore, since
,
the desired result is obtained:
.
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Product Rule
Consider the Product Rule:
.
Note that
,
for some value of
.
Rewriting the preceding logarithm in exponential form yields
.
Since
,
it can be stated that
and
,
for some values of
and
.
So,
can be rewritten as
.
Thus, by equating exponents (the technique for solving a Type I Exponential Equation),
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 Spring '14
 StephenFowler
 Algebra, Trigonometry, Derivative, exponential form yields

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