Exponents & Logarithms

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. \(\ln(x^2)=2\ln(x)\)