Exponents & Logarithms

Exponents are defined as:

• $b^n=\underset{n}{\underbrace{b\times b\times ...\times b}}$, where base $b$ is multiplied by itself $n$ times
• $b^0=1$ (for $b\ne 0$)

There are some common rules for exponents, assuming $x$ and $y$ are real numbers, $m$ and $n$ are integers, and $a$ and $b$ are rational:

1. $x^{-n}=\frac{1}{x^n}$
   • e.g. $x^{-3}=\frac{1}{x^3}$
2. $x^{\frac{1}{n}}=\sqrt[n]{x}$
   • e.g. $x^{\frac{1}{2}}=\sqrt{x}$
3. $x^{(\frac{m}{n})}=(x^{\frac{1}{n}}{)}^m$
   • e.g. $8^{\frac{4}{3}}=(8^{\frac{1}{3}}{)}^4=2^4=16$
4. $x^a{x}^b=x^{a+b}$
   • e.g. $x^2{x}^3=x^5$
5. $\frac{x^a}{x^b}=x^{a-b}$
   • e.g. $\frac{x^2}{x^3}=x^{-1}=\frac{1}{x}$
6. $(\frac{x}{y}{)}^a=\frac{x^a}{y^a}$
   • e.g. $(\frac{x}{y}{)}^2=\frac{x^2}{y^2}$
7. $(xy{)}^a=x^a{y}^a$
   • e.g. $(xy{)}^2=x^2{y}^2$

Logarithms are the exponents in the expressions above, the inverse of exponentiation

• If $b^y=x$, then $lo{g}_b(x)=y$
  • $y$ is the number you must raise $b$ to in order to get $x$
  • e.g. $2^6=64=\underset{\text{6 times}}{\underbrace{(2\ast 2\ast 2\ast 2\ast 2\ast 2)}}$ so $lo{g}_2(64)=6$

We often use the natural logarithm (ln) with base $e=2.718...$ in many math, statistics, and economic applications

• If $e^y=x$, then $\ln (x)=y$

There are a number of highly useful rules for natural logs:

1. $\ln (xy)=\ln (x)+\ln (y)$
   • e.g. $\ln (2\ast 3)=\ln (2)+\ln (3)$
2. $\ln (\frac{x}{y})=\ln (x)-\ln (y)$
   • e.g. $\ln (\frac{2}{3})=\ln (2)-\ln (3)$
3. $\ln (x^a)=a\ast \ln (x)$
   • e.g. $\ln (x^2)=2\ln (x)$