To determine the power function `y=ax^b` from the given coordinates: `(5,10)` and `(12,81)` , we set-up system of equations by plug-in the values of x and y on `y=ax^b.`

Using the coordinate `(5,10)`, we let x=5 and `y =10`.

First equation:` 10 = a*5^b`

Using the coordinate `(12,81)`, we let `x=12` and `y =81` .

Second equation: `81 = a*12^b`

Isolate "`a` " from the first equation.

`10 = a*5^b`

`10/5^b= (a*5^b)/5^b`

`a= 10/(5^b)`

Plug-in `a=10/5^b` on `81 = a*12^b` , we get:

`81=10/5^b*12^b`

`81 = 10*12^b/5^b`

`81 = 10*(12/5)^b`

`81/10 = (10*(12/5)^b)/10`

`81/10=(12/5)^b`

`8.1=(2.4)^b`

Take the "`ln` " on both sides to bring down the exponent by applying the natural logarithm property: `ln(x^n)=n*ln(x)` .

`ln(8.1)=ln(2.4^b)`

`ln(8.1)=b ln(2.4)`

Divide both sides by `ln(2.4)` to isolate `b` .

`(ln(8.1))/(ln(2.4))=(b ln(2.4))/(ln(2.4))`

`b =(ln(8.1))/(ln(2.4)) or 2.389 ` (approximated value)

Plug-in `b~~ 2.389` on `a=10/5^b` , we get:

`a=10/5^2.389`

`a~~0.214`

Plug-in `a~~0.214` and `b~~ 2.389` on `y=ax^b` , we get the power function as:

`y =0.214x^2.389`