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Exponent Laws & Rules

Introduction to exponentials

  • Exponents and logarithms are closely related. For example:
    \begin{eqnarray*}
    2^{3} &=& 8 \\
    \log_{2} 8 &=& 3
    \end{eqnarray*}
  • The general case is:
  • \begin{eqnarray*}
    a^{x}= y &\leftrightarrow &
    \log_{a} y = x \\
    e^{x} = y &\leftrightarrow& \ln
    y = x
    \end{eqnarray*}
  • An exponential equation represents exponential growth or decay.
    • An example of exponential growth is the compound interest formula
      \[ A = P \left(1+\frac{r}{100}\right)^{n}\]  where \(A\) is the amount after \(n\) periods, \(P\) is the principal invested (initial amount invested), and \(r\) is the interest rate per period of time.
    • Another example is radioactive decay:
      \[ M = M_{0} e^{-kt}\] where \(M\) is the mass after \(t\) years, if the initial mass was \(M_{0}\) and \(k\) is the decay factor.
  • In the expression, \(2^{3}\), the \(2\) is termed the base and \(3\) is the exponent.
  • The exponent tells us how many times to multiply the number by itself.
  • Example: \(\displaystyle 2^3 = 2 \times 2 \times 2 = 8\)