1. If a x = n(a >；; 0, and a≠ 1), then the number x is called the logarithm of n with a base, and it is denoted as x=logN, where a is called the base of logarithm and n is called a real number. And a>o, a≠ 1, N>0.
2. In particular, we call the logarithm based on 10 as the common logarithm, and log 10N as lgN.
3. Logarithms based on irrational number e (e=2.7 1828 ...) are called natural logarithms, and logeN is called lnN.
Zero has no logarithm.
In the range of real numbers, negative numbers have no logarithm. In the range of complex numbers, negative numbers have logarithm. For example:
㏑(-5)=㏑[(- 1)*5]=㏑(- 1)+㏑5=iπ+㏑5.
In fact, when θ=(2k+ 1)π (k∈Z), e [(2k+ 1) π i]+ 1 = 0, so ㏑(- 1) will be available. In this way, the natural logarithm of any negative number has periodic multivalues. For example: ㏑(-5)=(2k+ 1)πi+㏑5.
loga 1=0，logaa= 1
2 basic properties edit this paragraph
If a>0, and a≠ 1, m >；; 0, N>0, then:
1, logarithm (a) n = n (logarithmic identity)
Proof: let log(a) N=t, (t∈R)
Then there is a t = n.
a^(log(a)N)=a^t=N.
Prove [2]
2、log(a) a= 1
Certificate: because a b = a b
Let t = a b
So a b = t, b = log (a) (t) = log (a) (a b)
Let b= 1, then1= log (a) a.
3、log(a)(M ^ N)= log(a)M+log(a)N
Equation 54, log (a) (m÷ n) = log (a) m-log (a) n m-log (a) n.
M n = nlog (a meter
6、log(a)b*log(b)a= 1
7. log(a) b=log (c) b÷log (c) a (formula)
Basic Attribute 5 Promotion
log(a^n)(b^m)=m/n*[log(a)(b)]
Derived as follows:
Reciprocal formula
log(a^n)(b^m)=ln(b^m)÷ln(a^n)
Derivation of the formula for changing the bottom;
Let e x = b m and 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)
De: log (a n) (b m) = ln (b m) ÷ ln (a n)
Through basic attribute 5
log(a^n)(b^m) = [m×ln(b)]
Then it can be obtained from the formula of changing the bottom.
log(a^n)(b^m)=m÷n×[log(a)(b)]