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# Exponential Functions

## Exponential Functions 1.8 - Solution

We want to identify the value of the base of the exponential function from the given graph.

Let's consider the given equation for this graph. $\begin{gathered} f(x)=b^x \end{gathered}$ Let's first substitute the given point which has $1$ as its $x\text{-}$coordinate into the above formula. This will allow us to find the value of $b.$ Let's substitute $(1,5)$ into the above formula.
$f(x)=b^x$
${\color{#009600}{5}}=b^{\color{#0000FF}{1}}$
Solve for $b$
$5=b$
$b=5$
Now we can write our equation. $\begin{gathered} f(x)={\color{#FF0000}{b}}^x \quad \Rightarrow \quad f(x)={\color{#FF0000}{5}}^x \end{gathered}$ Next, we will substitute other given points, $(0,1)$ and $\left(\text{-} 1, \frac{1}{5}\right),$ into the above equation to check it. Let's start from $(0,1).$
$f(x)=5^x$
${\color{#009600}{1}}=5^{\color{#0000FF}{0}}$
$1=1$
The second point satisfies the equation. Let's try the last one, $\left(\text{-} 1, \frac{1}{5}\right).$
$f(x)=5^x$
${\color{#009600}{\dfrac{1}{5}}}=5^{\color{#0000FF}{\text{-} 1}}$
$\dfrac{1}{5}=\dfrac{1}{5^1}$
$\dfrac{1}{5}=\dfrac{1}{5}$
All the given points satisfy the equation. Now we can be sure that $b=5.$