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
Logarithm (a)(MN) = Logarithm (a)(M) Logarithm (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:
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)]
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)]