Ok it seems too much to cover.. anyway as posted in #2 already give you the algorithm.

You may try to do the following with your excel:

1. Transform the variable as follow: by taking log from the equation

\( y \triangleq \ln t_{0.5j}, x_1 \triangleq \ln S_{T_j}, x_2 \triangleq \frac {1} {8.31T_j} \)

2. Put the variables \( \mathbf{y} \) as a column vector.

Create a column vector \( \mathbf{1} \) match with your data size, along with the column vectors \( \mathbf{x}_1, \mathbf{x}_2 \). These 3 columns vectors combine to from the design matrix, \( \mathbf{X} \triangleq\begin{bmatrix} \mathbf{1} && \mathbf{x}_1 && \mathbf{x}_2 \end{bmatrix}\)

3. So for this multiple linear regression, \( y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon \)

the least square estimate is given by

\( \begin{bmatrix} \hat{\beta}_0 \\ \hat{\beta}_1 \\ \hat{\beta}_2 \end{bmatrix}= (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{Y} \)

The transpose and matrix inverse function is readily available in Excel.

If you want to understand why this is the solution, just need to do differentiation and solve the system of linear equations.

4. Transform the obtained estimates \( \hat{\beta}_0, \hat{\beta}_1, \hat{\beta}_2\) to \( \hat{A}_1, \hat{A}_2, \hat{A}_3 \)

You just need to write \( A_1, A_2, A_3 \) as the subjects, matching the transformation in equation 1.

I think the steps are quite clear now, and thats what I can help.

Of course this step just help you to do the regression manually in excel. If you understand everything, actually many statistical software, even in excel has built-in function for regression.