Can somebody please help me with the following problem.
I want to solve the following power function for the $\phi$ coefficient:
\begin{equation} \hat{z} = x\phi^\alpha \end{equation}
I have come up with two solutions.
Solution 1:
Step 1) \begin{equation} \frac{\hat{z}}{x} = \phi^\alpha \end{equation} Step 2) \begin{equation} \left(\frac{\hat{z}}{x}\right)^\frac{1}{\alpha} = \phi \end{equation}
Solution 2:
Step 1) \begin{equation} \log_{\phi}\left(\frac{\hat{z}}{x}\right) = \alpha \end{equation} Step 2) \begin{equation} \frac{\log_{\gamma}\left(\frac{\hat{z}}{x}\right)}{\log_{\gamma}\phi} = \alpha \end{equation} Step 3) \begin{equation} \frac{\log_{\gamma}\left(\frac{\hat{z}}{x}\right)}{\alpha} = \log_{\gamma}\phi \end{equation} Step 4) \begin{equation} \phi = \gamma^{\frac{\log{\gamma}\left(\frac{\hat{z}}{x}\right)}{\alpha}} \end{equation}
Plotting both functions in Matlab (with a log-log plot) shows that both functions are indeed different Plot of both functions taking 1->100 as the X-values. Plot constructed using MATLAB software
Can anyone please tell me which derivation is correct? The first one is simpler, but I believe I have applied the correct laws in the second solution too, yet I have arrived at different results.
$$z=x\phi^{\alpha} \implies \ln\frac{z}{x} = \alpha \ln \phi \implies \ln \phi = \frac{ \ln\frac{z}{x}}{\alpha} = \ln \left(\frac{z}{x}\right)^{\frac{1}{\alpha}} \implies \phi = \left(\frac{z}{x}\right)^{\frac{1}{\alpha}}$$
So, they are the same.