Question

The following data points are expected to follow a functional form of y = ax^b . Obtain the values of a and b by converting the functional form to a linear relationship between log(x) and log(y).

x 1.21 1.35 2.40 2.75 4.50 5.10 7.1 8.1
y 1.20 1.82 5.00 8.80 19.5 32.5 55.0 80.0

Answer 1

Explanation:

Given that,

y=ax^b

Taking the logarithm of both sides

Log(y) =Log(ax^b)

Applying the law of logarithm

Log(aⁿ) =nLog(a)

Therefore,

Log(y) =b Log(ax)

Also applying the product rule of logarithm

LogAB= Log A +Log B

Then,

Log(y) =b{Log(a) +Log(x)}

Log(y) =b•Log(x) +b•Log(a)

So, this is the linear relationship

Now,

Using the value given

When x=1.21, y=1.20

Then,

Log(y) =b{Log(a) +Log(x)}

Log(1.2) =b{Log(a) +Log(1.21)}

b= Log(1.2)/{Log(a) +Log(1.21)}

Also, when x=1.35, y=1.82

Then,

Log(1.82) =b{Log(a) +Log(1.35)}

Log(1.82) =b{Log(a) +Log(1.35)}

b= Log(1.82)/{Log(a) +Log(1.35)} equation 1

Equating the two b, since b is a constant

Log(1.2)/{Log(a) +Log(1.21)} = Log(1.82)/{Log(a) +Log(1.35)}

Cross multiply

Log(1.2)•{Log(a) +Log(1.35)} = Log(1.82)•{Log(a) +Log(1.21)}

Log(1.2)Log(a) + Log(1.2)log(1.35) = Log(1.82)Log(a) + Log(1.82)Log(1.21)

Collect like terms

Log(1.2)Log(a)-Log(1.82)Log(a) = Log(1.82)Log(1.21) - Log(1.2)log(1.35)

Log(a){Log(1.2)-Log(1.82)} = Log(1.82)Log(1.21)-Log(1.2)Log(1.35)

Log(a) = {Log(1.82)Log(1.21)-Log(1.2)Log(1.35)} / {Log(1.2)-Log(1.82)}

Then, Log(a)=0.01121/-0.18089

Log(a)=-0.06197

a=antilog(-0.06197)

a=0.867

Then, from equation 1

b= Log(1.82)/{Log(a) +Log(1.35)}

b= Log(1.82)/{Log(0.867) +Log(1.35)}

b=3.8

Then,

a=0.867 and

b=3.8

Therefore,

y=ax^b

y=0.867x^3.8