Exponents are defined as:

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

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. $$x^2x^3=x^5$$
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. $$(xy)^a=x^ay^a$$
• e.g. $$(xy)^2=x^2y^2$$

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

• If $$b^y=x$$, then $$log_b(x)= y$$
• $$y$$ is the number you must raise $$b$$ to in order to get $$x$$
• e.g. $$2^6=64 = \underbrace{(2*2*2*2*2*2)}_{\text{6 times}}$$ so $$log_2(64)=6$$

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