(5,10) , (12,81) 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: (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