I'm currently revising an exam about channel coding in telecommunications and we have a question where we need to isolate a variable $u$ in terms of another variable $q$. Currently, I am stuck with a rather difficult expression to simplify or at least I can't see what the trick is (if there is one). Here is the said expression:
$$log_2\big(q^q(1-q)^{1-q}\big) = log_2\bigg((\frac{qu}{1-qu})^q\bigg)$$
Being perfectly honest, I know that the $log_2$ simplifies on each side but having $u$ in both the numerator and denominator of the fraction which has an overall exponent, I'm quite out of my comfort zone mathematically.
Thanks in advance!
Since we have on both sides the base $2$ we get $$q^q(1-q)^{1-q}=\left(\frac {qu}{1-qu}\right)^q$$ Let's say $$A=q^q(1-q)^{1-q}$$ then $$a^{1/q}=\frac{qu}{1-qu}$$ so $$A^{1/q}-qA^{1/q}=qu$$ and $$A^{1/q}-quA^{1/q}=qu$$ so $$A^{1/q}=u(q+qA^{1/q})$$ and we obtain $$u=\frac{A^{1/q}}{q+qA^{1/q}}$$