(4,8) , (8,30) Write a power function y=ax^b whose graph passes through the given points

To determine the power function y=ax^b from the given coordinates: (4,8) and (8,30) , we set-up system of equations by plug-in the values of x and y on y=ax^b .
Using the coordinate (4,8) , we let x=4 and y =8 .
First equation: 8 = a*4^b
Using the coordinate (8,30) , we let x=8 and y =30 .
Second equation: 30 = a*8^b
Isolate "a" from the first equation.
8 = a*4^b
8/4^b= (a*4^b)/4^b
a= 8/4^b
Plug-in a=8/4^b on 30 = a*8^b , we get:
30 = 8/4^b*8^b
30 = 8*8^b/4^b
30 = 8*(8/4)^b
 30 = 8*(2)^b
30/8= (8*(2)^b)/8
15/4=2^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(15/4) =ln(2^b)
ln(15/4) =b*ln(2)
Divide both sides by ln(2) to isolate b.
(ln(15/4))/ln(2) =(b*ln(2))/(ln(2))
b =(ln(15/4))/ln(2) or 1.91 (approximated value).
Plug-in b= 1.91 on a=8/4^b , we get:
a=8/4^1.91
a~~ 0.566  (approximated value)
Plug-in a~~0.566 and b ~~ 1.91 on y =ax^b , we get the power function as:
y =0.566x^1.91