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## Homework Statement

Consider the system of equations

[tex]

\begin{eqnarray*}

e_1 &=& Ak^p + Bh^q \\

e_2 &=& A(k/2)^p + Bh^q \\

e_3 &=& Ak^p + B(h/2)^q \\

e_4 &=& A(k/2)^p + B(h/2)^q

\end{eqnarray*}

[/tex]

Suppose that the [itex]e_i[/itex] are known, as well as k and h. Find A, B, p, and q.

## Homework Equations

It's worth noting that a simpler case of this is

[tex]

\begin{eqnarray*}

e_1&=& Ak^p \\

e_2 &=& A(k/2)^p

\end{eqnarray*}[/tex]

Then to get the solution, you divide [itex]e_1/e_2 = 2^p[/itex], allowing you to solve for p.

## The Attempt at a Solution

I've tried a bunch of different combinations of the [itex]e_i[/itex]'s, but I always seem to have something like [itex]Ak^p(1-1/2^p)[/itex] attached to each other, preventing me from solving for one. For example,

[tex]

\begin{eqnarray*}

e_1-e_2 &=&Ak^p(1-1/2^p) \\

&=&e_3-e_4 \\

e_2-e_4 & = & Bh^q(1-1/2^q) \\

&=&e_1-e_3 \\

(e_1-e_2)(e_2-e_4)&=& Ak^pBh^q(1-1/2^p)(1-1/2^q) \\

p &=& \log_2\frac{e_1-Bh^q}{e_2-Bh^q}

\end{eqnarray*}

[/tex]

So I basically keep getting expressions that are useless, since you can't really solve for one in terms of the others. Does anyone have any ideas or experience with this sort of thing? Thanks in advance!