Laws of Exponents Review Sheet

Summary of Laws of Exponents and Log

Laws of Exponents

From the humble beginnings of the definition for integers:

{$$ \begin{align} b^1 &= b \cr b^{n+1} &= b^n b \end{align} $$}

the Laws of exponents can be (and in algebra are) derived, being

{$$ \begin{align} b^1 &= b \tag{i.. def.} \cr b^0 &= 1 \tag{ii.} \cr b^{-1} &= {1 \over b} \tag{iii.} \cr b^mb^n &= b^{m+n} \tag{iv.} \cr {{b^m} \over {b^n}} &= b^{m-n} \tag{v.} \cr (b^m)^n &= b^{mn} \tag{vi.} \cr b^{-n} &= {1 \over {b^n}} \tag{vii.} \cr (ab)^{n} &= a^nb^n \tag{viii.} \cr \left( {a \over b} \right)^{n} &= {{a^n} \over {b^n}} \tag{ix.} \cr \color{red}{\text{Warning!}} \cr \color{red}{\left( a+b \right)^n} & \color{red}{\ne a^n+b^n} \cr \color{red}{ a ^{n^m}} & \color{red}{\ne a^{nm}} \cr \end{align} $$}

These rules are extended by continuity to all real exponents. Note that a real number raised to any power is never zero.

Laws of Logarithms

In light of the definition of logarithms and

{$$ \begin{gather} \text{for } b \ge 0\text{, } b \ne 1 \text{, } \text{ and } y \gt 0 \text{, } \cr y=b^x \iff x = \log_b y \end{gather} $$}

(vi.) and (v.) give us:

{$$ \begin{align} \log_b(xy) &= \log_b x + \log_b y \tag{1.} \cr \log_b \left( {x \over y} \right) &= \log_b x - \log_b y \tag{2.} \cr \log_b x^r &= r \log_b x \tag{3.} \cr \log_b b^x &= x \tag{4., def.} \cr b^{\log_b x} &= x \tag{5., def.} \end{align} $$}



Categories: Algebra Topics, PreCalculus Topics



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