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: \(\displaystyle 2^3 = 2 \times 2 \times 2 = 8\)
To do
- Variable expressions with exponents activity (Khan Academy)
More info
- Introduction to Exponentials video (Study Support, USQ Library)
- Powers and roots quick reference (mathcentre)
- Indices or powers quick reference (mathcentre)
- The exponential constant e quick reference (mathcentre)