(3,1) , (5,4) Write an exponential function y=ab^x whose graph passes through the given points.

To determine the power function y=ax^b from the given coordinates: (3,1) and (5,4) , we set-up system of equations by plug-in the values of x and y on y=ax^b .
Using the coordinate (3,1) , we let x=3 and y =1 .
First equation: 1 = a*3^b
Using the coordinate (5,4) , we let x=5 and y =4 .
Second equation: 4 = a*5^b
Isolate "a " from the first equation.
1 = a*3^b
1/3^b= (a*3^b)/3^b
a= 1/3^b
Plug-in a=1/3^b on 4 = a*5^b , we get:
4 = 1/3^b*5^b
4= 1*5^b/3^b
4= 1*(5/3)^b
4= (5/3)^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(4)=ln((5/3)^b)
ln(4)=b ln(5/3)
Divide both sides by ln(5/3) to isolate b.
(ln(4))/(ln(5/3))=(b ln(5/3))/(ln(5/3))
b =(ln(4))/(ln(5/3)) or 2.714 (approximated value)
Plug-in b~~ 2.714 on a=1/3^b , we get:
a=1/3^2.714
a~~0.051
Plug-in a~~0.051 and b~~2.714 on y=ax^b , we get the power function as:
y =0.051x^2.714