Contact The Learning Centre

# 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: $$\displaystyle 2^3 = 2 \times 2 \times 2 = 8$$