Laws of Exponents & Logarithms

Exponents (powers or indices) are a shorthand for repeated multiplication: $a^n$ means $a$ multiplied by itself $n$ times. From this definition flow the laws of exponents, which hold for any real base $a > 0$ and real exponents $m, n$: $$a^m \cdot a^n = a^{m+n}, \qquad \frac{a^m}{a^n} = a^{m-n}, \qquad (a^m)^n = a^{mn}.$$ Products of powers with the same base add exponents; quotients subtract; a power raised to a power multiplies. Three further conventions extend the definition beyond the counting numbers: $a^0 = 1$ (so the pattern of dividing by $a$ continues), $a^{-n} = \dfrac{1}{a^n}$ (a negative exponent means reciprocal), and $a^{1/n} = \sqrt[n]{a}$ with $a^{m/n} = \sqrt[n]{a^m}$ (fractional exponents are roots). There are also distribution rules across a product or quotient: $(ab)^n