yoda78
11.03.2020 •
Mathematics
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
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Ответ:
Step-by-step 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
Ответ:
5x = 24
Step-by-step explanation:
Combine like terms
Further Workings :
Simplify