A friend of mine sent me his homework in economics since I put myself forward for solving it. One of the exercises was formulated as follows: $$ x = x(r_1,r_2) = 8 * r_1^\alpha *r_2^\beta$$ Knowing that: $ x(14400,32768) = 61440 $ and $ x(5184,7776) = 20736$ find the values of parameters $\alpha,\beta$.
My approach to the problem consists in applying the natural logarithm to both equations and then solving them as linear ones for the parameters. Since the numbers were cumbersome to work with i denoted them with capital letters so it came up to: $$ln(E) = ln(8)+\alpha ln(A)+\beta ln(B)$$ $$ln(F) = ln(8)+\alpha ln(C)+\beta ln(D)$$ I tried to solve it by multiplying the first equation by $log_BD$ but it turned out to be a real headache. After some calculations I came up with: $$\alpha = \frac{ln(\frac{F}{8(\frac{E}{8})^{log_BD}})}{ln(\frac{C}{A^{log_BD}})}$$ The exercise gave the opportunity of checking the results that were $\alpha =0.5, \beta=0.4$, therefore invalidating my answer as a longshot. What am I doing wrong?
