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