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I need to get $c$ in terms of $a$.

Both $c, a\in \Bbb R^+$ $$ c^{2a} = c/a $$

I'm pretty sure that log is required to solve this, but I'm not quite sure how to approach it.

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  • $\begingroup$ How about dividing by $c$ then taking the appropriate power? $\endgroup$ – Jonathan Y. Aug 29 '13 at 20:43
  • $\begingroup$ @JonathanY. Oh, that seems like it might make more sense. It just seemed to me like log would be needed, because of the powers $\endgroup$ – Alex Aug 29 '13 at 20:46
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Since $c>0$, we may divide both sides by $c$ to obtain: $$ c^{2a-1} = \frac1a $$ To undo the exponent, we raise both sides of the equation to the power of $\dfrac{1}{2a-1}$ to obtain: $$ c= \left(\frac1a\right)^{\dfrac{1}{2a-1}} $$

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    $\begingroup$ It should probably be pointed out that this doesn't work if $a=\frac{1}{2}$; however, in that case the equation is $c=c/a$, so that either $a=1$ or $c=0$. $\endgroup$ – Nick Peterson Aug 29 '13 at 20:46
  • $\begingroup$ @NicholasR.Peterson If $a=\frac{1}{2}$, how can $a=1$? $\endgroup$ – Ryan Aug 29 '13 at 22:07
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    $\begingroup$ Well, it can't. So that tells you that $c=0$. :-P $\endgroup$ – Nick Peterson Aug 29 '13 at 22:08

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