Untitled Flashcard Set
1. Basic Properties (Numbers & Operations)
Commutative Property
a+b=b+aa + b = b + aa+b=b+a or ab=baab = baab=ba
Associative Property
(a+b)+c=a+(b+c)(a + b) + c = a + (b + c)(a+b)+c=a+(b+c) or (ab)c=a(bc)(ab)c = a(bc)(ab)c=a(bc)
Distributive Property
a(b+c)=ab+aca(b + c) = ab + aca(b+c)=ab+ac
Identity Property
a+0=aa + 0 = aa+0=a or a⋅1=aa \cdot 1 = aa⋅1=a
Inverse Property
a+(−a)=0a + (-a) = 0a+(−a)=0 or a⋅1a=1a \cdot \frac{1}{a} = 1a⋅a1=1 (for a≠0a \neq 0a=0)
2. Algebraic Properties (Equality & Operations)
Addition Property of Equality
If a=ba = ba=b, then a+c=b+ca + c = b + ca+c=b+c
Multiplication Property of Equality
If a=ba = ba=b, then ac=bcac = bcac=bc
3. Exponent Properties
Product Rule
xm⋅xn=xm+nx^m \cdot x^n = x^{m+n}xm⋅xn=xm+n
Quotient Rule
xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}xnxm=xm−n
Power Rule
(xm)n=xmn(x^m)^n = x^{mn}(xm)n=xmn
Power of a Product Rule
(ab)n=anbn(ab)^n = a^n b^n(ab)n=anbn
Power of a Quotient Rule
(ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}(ba)n=bnan
Zero Exponent Rule
a0=1a^0 = 1a0=1 (for a≠0a \neq 0a=0)
Negative Exponent Rule
a−n=1ana^{-n} = \frac{1}{a^n}a−n=an1
Fractional Exponent Rule
am/n=amn=(an)ma^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^mam/n=nam=(na)m
4. Logarithmic Properties
Definition of a Logarithm
loga(b)=c ⟺ ac=b\log_a(b) = c \iff a^c = bloga(b)=c⟺ac=b
Log Product Rule
loga(MN)=loga(M)+loga(N)\log_a(MN) = \log_a(M) + \log_a(N)loga(MN)=loga(M)+loga(N)
Log Quotient Rule
loga(MN)=loga(M)−loga(N)\log_a\left(\frac{M}{N}\right) = \log_a(M) - \log_a(N)loga(NM)=loga(M)−loga(N)
Log Power Rule
loga(Mk)=k⋅loga(M)\log_a(M^k) = k \cdot \log_a(M)loga(Mk)=k⋅loga(M)
Change of Base Formula
loga(M)=logb(M)logb(a)\log_a(M) = \frac{\log_b(M)}{\log_b(a)}loga(M)=logb(a)logb(M)
Special Log Values / Identity
loga(a)=1\log_a(a) = 1loga(a)=1, loga(1)=0\log_a(1) = 0loga(1)=0