Answer: a = 47.3;
p = 0.65
Step-by-step explanation: A power function is of the form [tex]y=ax^{p}[/tex] and passes through points (2,5) and (3,81), i.e.:
[tex]5=a.2^{p}[/tex]
[tex]81=a.3^{p}[/tex]
To determine the two unknows, solve the system of equations:
[tex]log5=log(a.2^{p})[/tex]
[tex]log81=log(a.3^{p})[/tex]
Using multiplication and power rules:
[tex]log5=loga+plog2[/tex]
[tex]log81=loga+plog3[/tex]
Giving values to constants:
0.7=loga+0.3p
2=loga+0.5p
This system of equations can be solved by subtracting each other:
[tex]-1.3=-0.2p[/tex]
p = 0.65
Substituting p into one of the equations above:
[tex]loga+0.5(0.65)=2[/tex]
[tex]loga=1.675[/tex]
[tex]a=10^{1.675}[/tex]
a = 47.3
The constants a and p of the power function which passes through points (2,5) and (3,81) are 47.3 and 0.65, respectively.
