Exponential Functions and the Number e

Starting point: Years 9 and 10

Index laws reminder

$$a^m{a}^n=a^{m+n}\frac{a^m}{a^n}=a^{m-n}(a^m{)}^n=a^{mn}{a}^0=1a^{-n}=\frac{1}{a^n}$$

Exponential functions

In an exponential function, the variable is in the power:

$$y=a^x(a>0,a\ne 1)$$

$y={\left(\frac{1}{2}\right)}^x=2^{-x}$ is the mirror image in the y-axis. It shows decay.

Growth and decay

Something that grows by a fixed percentage follows $A=P(1+r{)}^n$. Something that shrinks by a fixed percentage follows $A=P(1-r{)}^n$.

Solving simple exponential equations

Write both sides with the same base, then match the powers.

$$3^{x+1}=81=3^4\Rightarrow x+1=4\Rightarrow x=3$$

Quick summary

• In an exponential function the variable is the power: $y=a^x$.
• $y=a^x$ passes through $(0,1)$ with asymptote $y=0$. $a>1$ grows, $0<a<1$ decays.
• Percentage growth: $A=P(1+r{)}^n$. Percentage decay: $A=P(1-r{)}^n$.
• Solve $a^{f(x)}=a^k$ by matching powers.
• $e\approx 2.718$ and $\frac{d}{dx}{e}^x=e^x$.

Quiz