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: \(\displaystyle 2^3 = 2 \times 2 \times 2 = 8\)