The proof of the arithmetic of logarithm function in the first grade of senior high school

The proof of the arithmetic of logarithm function in the first grade of senior high school

Arithmetic of logarithm function in the first year of senior high school
1. A ^ (log (a) (b)) = B (logarithmic identity)
　　2、log(a)(a^b)=b
　　3、log(a)(MN)=log(a)(M)+log(a)(N);
　　4、log(a)(M÷N)=log(a)(M)-log(a)(N);
　　5、log(a)(M^n)=nlog(a)(M)
　　6、log(a^n)M=1/nlog(a)(M)
Certification:
1. Because n = log (a) (b), substituting a ^ n = B, that is, a ^ (log (a) (b)) = B
2. Because a ^ B = a ^ B
Let t = a ^ B
So a ^ B = t, B = log (a) (T) = log (a) (a ^ b)
　　3、MN=M×N
From the basic property 1 (replace m and N)
　　a^[log(a)(MN)] = a^[log(a)(M)]×a^[log(a)(N)] =(M)*(N)
By the nature of index
　　a^[log(a)(MN)] = a^{[log(a)(M)] + [log(a)(N)]}
The two methods are only different in nature, and the method adopted depends on the actual situation
And because exponential function is monotone function, so
　　log(a)(MN) = log(a)(M) + log(a)(N)
4. Similar to (3)
　　MN=M÷N
From the basic property 1 (replace m and N)
　　a^[log(a)(M÷N)] = a^[log(a)(M)]÷a^[log(a)(N)]
By the nature of index
　　a^[log(a)(M÷N)] = a^{[log(a)(M)] - [log(a)(N)]}
And because exponential function is monotone function, so
　　log(a)(M÷N) = log(a)(M) - log(a)(N)
5. Similar to (3)
　　M^n=M^n
From basic property 1 (replace m)
　　a^[log(a)(M^n)] = {a^[log(a)(M)]}^n
By the nature of 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)
Basic properties 4 generalization
　　log(a^n)(b^m)=m/n*[log(a)(b)]
The derivation is as follows
According to the bottom changing formula (see below) [LNX is log (E) (x), e is called the bottom of natural logarithm]
　　log(a^n)(b^m)=ln(b^m)÷ln(a^n)
The derivation of the formula is as follows
Let e ^ x = B ^ m, e ^ y = a ^ n
Then log (a ^ n) (b ^ m) = log (e ^ y) (e ^ x) = x / Y
　　x=ln(b^m),y=ln(a^n)
Get: log (a ^ n) (b ^ m) = ln (b ^ m) △ ln (a ^ n)
From the basic properties 4
　　log(a^n)(b^m) = [m×ln(b)]÷[n×ln(a)] = (m÷n)×{[ln(b)]÷[ln(a)]}
And then by changing the bottom formula
　　log(a^n)(b^m)=m÷n×[log(a)(b)]
For example: log (8) 27 = log (2 & # 179;) 3 & # 179; = log (2) 3
Another example: log (√ 2) √ 5 = log (2) 5