# 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.

- An example of exponential growth is the compound interest formula
- 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: \(\displaystyle 2^3 = 2 \times 2 \times 2 = 8\)