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Properties and Derivation of Logarithm

Use 0 to represent the power, and log(a)(b) to represent the logarithm of b with a as the base.

* means multiplication symbol,/means division symbol.

Define formula:

If a n = b(a >;; 0 and a ≠ 1)

Then n=log(a)(b)

Basic nature:

1.a^(log(a)(b))=b

2 . log(a)(MN)= log(a)(M)+log(a)(N);

3 . log(a)(M/N)= log(a)(M)-log(a)(N);

4.log(a)(M^n)=nlog(a)(M)

infer

1. This need not be pushed, but can be obtained directly from the definition (bring [n=log(a)(b)] in the definition into a n = b).

2.

MN=M*N

By the basic properties of 1 (replacing m and n)

a^[log(a)(mn)]= a^[log(a)(m)]* a^[log(a)(n)]

According to the nature of the index

a^[log(a)(mn)]= a^{[log(a)(m)]+[log(a)(n)]]

And because exponential function is monotone function, so

log(a)(MN) = log(a)(M) + log(a)(N)

3. Similar to 2.

MN=M/N

By the basic properties of 1 (replacing m and n)

a^[log(a)(m/n)]= a^[log(a)(m)]/a^[log(a)(n)]

According to the nature of the index

a^[log(a)(m/n)]= a^{[log(a)(m)]-[log(a)(n)]]

And because exponential function is monotone function, so

Logarithm (a)(M/N) = Logarithm (a)(M)-Logarithm (a)(N)

4. Similar to 2.

M^n=M^n

From the basic attribute 1 (replace m)

a^[log(a)(m^n)]= {a^[log(a)(m)]}^n

According to the nature of the index

a^[log(a)(m^n)]= a^{[log(a)(m)]*n}

And because exponential function is monotone function, so

log(a)(M^n)=nlog(a)(M)

Other attributes:

Attribute 1: bottoming formula

log(a)(N)=log(b)(N) / log(b)(a)

Derived as follows

N = a^[log(a)(N)]

a = b^[log(b)(a)]

By combining the two formulas, it can be concluded that.

n = {b^[log(b)(a)]}^[log(a)(n)]= b^{[log(a)(n)]*[log(b)(a)]}

And because n = b [log (b) (n)]

therefore

b^[log(b)(n)]= b^{[log(a)(n)]*[log(b)(a)]}

therefore

log(b)(n)=[log(a)(n)]*[log(b)(a)]

So log(a)(N)=log(b)(N)/log(b)(a)

Nature 2: (I don't know what it's called)

log(a^n)(b^m)=m/n*[log(a)(b)]

Derived as follows

Through the formula [lnx is log (e) (x), and e is called the base of natural logarithm]

log(a^n)(b^m)=ln(a^n) / ln(b^n)

It can be obtained from basic attribute 4.

log(a^n)(b^m)=[n * ln(a)]/[m * ln(b)]=(m/n)* {[ln(a)]/[ln(b)]

Then according to the bottom changing formula

log(a^n)(b^m)=m/n*[log(a)(b)]

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Formula 3:

log(a)(b)= 1/log(b)(a)

Proved as follows:

Log(a)(b)= log(b)(b)/log(b)(a)- Logarithm based on b, log(b)(b)= 1.

= 1/log(b)(a)

Also deformable:

log(a)(b)*log(b)(a)= 1

Sum and difference product formula of trigonometric function

sinα+sinβ= 2 sin(α+β)/2 cos(α-β)/2

sinα-sinβ= 2cos(α+β)/2 sin(α-β)/2

cosα+cosβ= 2 cos(α+β)/2 cos(α-β)/2

cosα-cosβ=-2 sin(α+β)/2 sin(α-β)/2

Formula of product and difference of trigonometric function

sinαcosβ= 1/2[sin(α+β)+sin(α-β)]

cosαsinβ= 1/2[sin(α+β)-sin(α-β)]

cosαcosβ= 1/2[cos(α+β)+cos(α-β)]

sinαsinβ=- 1/2[cos(α+β)-cos(α-β)]

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