Challenge: Suppose you know the initial dose of a drug is 500 mg, and 3 hours later, there are 325 mg left, but you don't know the base multiplier [tex]\(b\)[/tex]. Use this information in the equation [tex]y = a \cdot b^x[/tex] to solve for [tex]\(b\)[/tex].

First, substitute [tex]\(a = 500\)[/tex], and use [tex]\(x = 3\)[/tex] with [tex]\(y = 325\)[/tex].

The equation will look like:
[tex]\[ 325 = 500 \cdot b^3 \][/tex]

Now solve for [tex]\(b\)[/tex]:

First, divide both sides by 500:
[tex]\[ \frac{325}{500} = b^3 \][/tex]
[tex]\[ 0.65 = b^3 \][/tex]

To undo a third power, take the cube root of both sides:
[tex]\[ \sqrt[3]{0.65} = \sqrt[3]{b^3} \][/tex]
[tex]\[ 0.866 = b \][/tex]

If your calculator doesn't have a cube root function, you can raise both sides of the equation to the [tex]\(\frac{1}{3}\)[/tex] power:
[tex]\[ (0.65)^{\frac{1}{3}} = \left(b^3\right)^{\frac{1}{3}} \][/tex]
[tex]\[ 0.866 = b \][/tex]

So the equation for this problem is:
[tex]\[ y = 500(0.866)^x \][/tex]