## Related questions with answers

The article “Mechanistic-Empirical Design of Bituminous Roads: An Indian Perspective” presents an equation of the form $y=a\left(1 / x_{1}\right)^{b}\left(1 / x_{2}\right)^{c}$ for predicting the number of repetitions for laboratory fatigue failure (y) in terms of the tensile strain at the bottom of the bituminous beam $(x_1)$ and the resilient modulus $(x_2).$ Transform this equation into a linear model, and express the linear model coefficients in terms of a, b, and c.

Solution

VerifiedGiven:

$y=a\left(\dfrac{1}{x_1}\right)^b \left(\dfrac{1}{x_2}\right)^c$

Rewrite the model as the product of powers of $x_i$:

$y=ax_1^{-b}x_2^{-c}$

Take the natural logarithm of each side of the previous equation:

$\ln y = \ln (a x_1^{-b}x_2^{-c})$

The logarithm of a product is the sum of the logarithms:

$\ln y = \ln a+\ln x_1^{-b}+\ln x_2^{-c}$

The power of a value in a logarithm can be moved to right before the logarithm (power property of logarithms).

$\ln y = \ln a-b\ln x_1-c\ln x_2$

Finally, we include a possible error $\epsilon$ which can occur in any predicted model:

$\ln y = \ln a-b\ln x_1-c\ln x_2 +\epsilon$

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