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`