Question

Write an exponential function in the form y=ab^x that goes through points (0,10) and (9, 5120).

Answer 1

The form
`y=ab^x` has two unknowns, namely
`a` and
`b`, so you would need to have two points to find the equation (which you do). Once you have your two points, you can create a system of equations by plugging in the x and y coordinates like below:

`\left \{ {{10=ab^0} \atop {5120=ab^9}} \right.`

One thing you may remember is that ANY number to the power of 0 equals 1. Knowing that, we can replace
`b^0` in the first equation with 1. Since 1 multiplied by any number is just that same number, that leaves
`a` alone in the first equation.

`\left \{ {{10=a} \atop {5120=ab^9}} \right.`

The first equation now tells us that
`a` is 10, so we can plug 10 in for
`a` in the other equation.

`5120=10b^9`

Now we can divide both sides by 10 to isolate
`b^9`.

`512=b^9`

To get
`b` alone, we have to take the 9th root of both sides.

`\sqrt[9]{512} =\sqrt[9]{b^9}`

That gives us:

`2=b`

We can now plug
`a` and
`b` into the form they gave us to get:

`y=10*2^x`