Given:
Power function and point is
Points
[tex]\begin{gathered} (1,4) \\ (5,41)^ \end{gathered}[/tex]
Find-:
Power of function
Explanation-:
Let the power function is:
[tex]f(x)=ax^n[/tex]
Check for points (1,4)
[tex]\begin{gathered} f(x)=ax^n \\ \\ (x,f(x))=(1,4) \\ \\ 4=a(1)^n \\ \\ a=4..................(1) \end{gathered}[/tex]
Check for points (5,41)
[tex]\begin{gathered} f(x)=ax^n \\ \\ f(x)=4x^n \\ \\ (x,f(x))=(5,41) \end{gathered}[/tex]
Put the value then,
[tex]\begin{gathered} f(x)=4x^n \\ \\ 41=4(5)^n \\ \\ \frac{41}{4}=5^n \\ \\ 10.25=5^n \end{gathered}[/tex]
Taking log both sides then
[tex]\begin{gathered} 10.25=5^n \\ \\ \ln(10.25)=\ln5^n \\ \\ \ln(10.25)=n\ln5 \\ \\ 2.327=n\times1.609 \\ \\ n=\frac{2.327}{1.609} \\ \\ n=1.446 \end{gathered}[/tex]
So the function is:
[tex]\begin{gathered} y=ax^n \\ \\ y=4x^{1.446} \end{gathered}[/tex]
