Exponential functions
An exponential function is a function in which the variable appears in the exponent. On this page we look at what an exponential function is, how its graph looks, and how to solve exponential equations.
Exponential function
An exponential function has the form
$f(x)=b^{x}$
where the base $b$ is a positive number with $b\ne 1$, and the variable $x$ is in the exponent. The following are exponential functions:
$f(x)=2^{x}$, $g(x)=3(1.5)^{x}$, $p(t)=4e^{t/2}$, $r(t)=4e^{-5t}$
By contrast, $f(x)=5x+4$ is not an exponential function, because the variable is not in the exponent.
Graphs of exponential functions
To graph an exponential function, make a table of values and plot the points. If the base is greater than $1$, the graph rises — this is exponential growth. If the base is between $0$ and $1$ (or the exponent is negative), the graph falls — this is exponential decay. Every exponential function $f(x)=b^{x}$ passes through $(0,1)$, because $b^{0}=1$.
Example 1: Graph the function $f(x)=3^{x}$.
Solution: Make a table of values:
| $x$ | $y=3^{x}$ |
|---|---|
| $-2$ | $3^{-2}=\dfrac{1}{9}$ |
| $-1$ | $3^{-1}=\dfrac{1}{3}$ |
| $0$ | $3^{0}=1$ |
| $1$ | $3^{1}=3$ |
| $2$ | $3^{2}=9$ |
Plotting these points and joining them with a smooth curve gives an increasing (growth) curve:
Example 2: Graph the function $f(x)=2^{-x}$.
Solution: Make a table of values:
| $x$ | $y=2^{-x}$ |
|---|---|
| $-2$ | $2^{2}=4$ |
| $-1$ | $2^{1}=2$ |
| $0$ | $2^{0}=1$ |
| $1$ | $2^{-1}=\dfrac{1}{2}$ |
| $2$ | $2^{-2}=\dfrac{1}{4}$ |
Plotting these points gives a decreasing (decay) curve:
Solving exponential equations
An exponential equation has the variable in an exponent. One way to solve such an equation is to write both sides as powers of the same base. If the bases are equal, then the exponents must be equal:
if $b^{m}=b^{n}$, then $m=n$.
Example 1: Solve $2^{x}=64$.
Solution: Write $64$ as a power of $2$: $64=2^{6}$. So
$2^{x}=2^{6}$
The bases are the same, so the exponents are equal:
$x=6$
Example 2: Solve $25^{x}=125$.
Solution: Write both sides as powers of $5$: $25=5^{2}$ and $125=5^{3}$:
$5^{2x}=5^{3}$
Set the exponents equal and solve:
$\begin{align*}2x&=3\\x&=\dfrac{3}{2}\end{align*}$
Example 3: Solve $27^{5x+4}=81^{4x-6}$.
Solution: Write both sides as powers of $3$: $27=3^{3}$ and $81=3^{4}$:
$3^{3(5x+4)}=3^{4(4x-6)}$
Set the exponents equal and solve:
$\begin{align*}3(5x+4)&=4(4x-6)\\15x+12&=16x-24\\12+24&=16x-15x\\36&=x\end{align*}$
So $x=36$.
Example 4: Solve $4^{x}=\dfrac{1}{1024}$.
Solution: Since $1024=4^{5}$, we have $\dfrac{1}{1024}=4^{-5}$:
$4^{x}=4^{-5}$
The bases are the same, so
$x=-5$
Example 5: Solve $20^{x}=0.05$.
Solution: Write $0.05$ as a power of $20$: $0.05=\dfrac{1}{20}=20^{-1}$:
$20^{x}=20^{-1}$
The bases are the same, so
$x=-1$