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Laws for positive integral indices

If a, b are real numbers and m, n are positive integers, we have the following laws:

(a)a^m×aⁿ=a^m+n

(b)a^m∫aⁿ=a^m-n

(c)(a^m)ⁿ=a^mn

(d)a^m×b^m=(ab)^m

(e)(a/b)^m=(a^m)/b^m

(f)(-1)ⁿ=-1, when n=2m-1; =1, when n=2m, where m is an integer.

Laws for fractional, zero and negative indices

(a)a⁰=1(a is not equal to 0)

(b)a^-m=1/(a^m)(a is not equal to 0)

(c)(≡a²)=|a| = +a, if a is bigger or equal to 0; =-a, if a is smaller than 0.

癸计猭玥

Definition: If a^x=N, then x=log_aN

       *[a is bigger than 0 and not equal to 1, N is bigger than 0]

(a)log_a1=0

(b)log_aa=1

(c)log_aM+log_aN=log_aMN

(d)log_aM-log_aN=log_a(M/N)

(e)log_aM^p=p log_aM

(f)log_aM^1/r=(1/r)×log_aM

(g)log_aM=(log_bM)/(log_ba)

(h)a^logaN=N

(i)log_a(MN/XY)=log_aM+log_aN-log_aX-log_aY