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The Change of Base Formula
When a,b>0, a≠1, for any c>0, c≠1
(Proof)logab=
(Review)
LetThe Definition of The Logarithm logay=x ←→ y=ax (y>0, a>0, a≠1) The Power of the Logarithm logaMn=n logaM logab=x …(A)(b>0, a>0, a≠1) then b=ax …(B) Take logarithm on both sides of (B) with a base c (c>0, c≠1) logcb=logcax logcb=x logca ← The Power of the Logarithm logcb=logab · logca ← (A) Therefore, logab= |
Example (1)log23=== (2)log84==
”To take a logarithm” doesn’t mean ”to remove a logarithm”,
logab → b but means ”to think of a logarithm”. b → logcb (with a base c) ax → logcax (with a base c) The following Rules of the Loagarithm work if and only if the bases are the same.
1.logaM+logaN=logaMN
When the logarithms have different bases, in order to use these rules you should rewrite the logarithms with the same base.
2.logaM−logaN=loga 3.n logaM=loga(M n)=logaM n |
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ExampleSimplify the following logarithms.
(1)log23·log32
(Answer 1)log23·log32=log23 =log22=1 (Answer 2) log23·log32= =1
(2)log56·log67·log75
(Answer)log56·log67·log75= =1 |
(3)log84
(Answer)log84====
(4)log9
(Answer)log9====−2 |
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Question 1Simplify the following logarithms. (Fill in the blanks.) |
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If the base of a logarithm and the base of an exponent are the same,
The logarithm with exponential argument is itself.
logaax=x …(I) The exponent with a logarithmic argument is itself. alogax=x …(II) |
(Proof) I ← logaax=x logaa=x II ← logax=logax ← self-evident truth According to the Definition of the Logarithm: y=logax → ay=x logax=logax → alogax=x |
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Question 2Simplify the following logarithms. (Fill in the blanks.) |