当a>0且a≠1时,M>0,N>0,那么: (1)log(a)(MN)=log(a)(M)+log(a)(N); (2)log(a)(M/N)=log(a)(M)-log(a)(N); (3)log(a)(M^n)=nlog(a)(M) (n∈R) (4)log(a^n)(M)=1/nlog(a)(M)(n∈R) (5)换底公式:log(A)M=log(b)M/log(b)A (b>0且b≠1) (6)a^(log(b)n)=n^(log(b)a) 证明: 设a=n^x则a^(log(b)n)=(n^x)^log(b)n=n^(x·log(b)n)=n^log(b)(n^x)=n^(log(b)a) (7)对数恒等式:a^log(a)N=N; log(a)a^b=b (8)由幂的对数的运算性质可得(推导公式) 1.log(a)M^(1/n)=(1/n)log(a)M ,log(a)M^(-1/n)=(-1/n)log(a)M 2.log(a)M^(m/n)=(m/n)log(a)M ,log(a)M^(-m/n)=(-m/n)log(a)M 3.log(a^n)M^n=log(a)M ,log(a^n)M^m=(m/n)log(a)M 4.log(以 n次根号下的a 为底)(以 n次根号下的M 为真数)=log(a)M , log(以 n次根号下的a 为底)(以 m次根号下的M 为真数)=(n/m)log(a)M 5.log(a)b×log(b)c×log(c)a=1
对数与指数之间的关系
当a>0且a≠1时,a^x=N x=㏒(a)N
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