當a>0且a≠1時,M>0,N>0,那麼：（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）換底公式：log(A)M=log(b)M/log(b)A (b>0且b≠1）(5) a^(log(b)n)=n^(log(b)a) 證明：設a=n^x 則a^(log(b)n)=（n^x）^log(b)n=n^（x·log(b)n）=n^log(b)（n^x）=n^(log(b)a)(6)對數恆等式：a^log（a)N=N;log（a）a^b=b（7）由冪的對數的運算性質可得(推導公式)1.log(a)M^(1/n)=(1/n)log(a)M ,log(a)M^(-1/n)=(-1/n)log(a)M2.log(a)M^(m/n)=(m/n)log(a)M ,log(a)M^(-m/n)=(-m/n)log(a)M3.log(a^n)M^n=log(a)M ,log(a^n)M^m=(m/n)log(a)M4.log(以 n次根號下的a 為底)(以 n次根號下的M 為真數)=log(a)M ,log(以 n次根號下的a 為底)(以 m次根號下的M 為真數)=(m/n)log(a)M5.log(a)b×log(b)c×log(c)a=1對數與指數之間的關係當a>0且a≠1時,a^x=N x=㏒(a)N