# Solving $ab-c+2^x=2^xd$ for $x$

I am trying to solve for $$x$$ in this equation: $$ab-c+2^x=2^xd$$ I've been trying to use Maple to figure it out but to no avail.

Looking for the mathematical solution and if possible a way to get the step-by-step solution in Maple also.

• Subtract $2^x$ from both sides, and divide both sides by $d-1$? Then take a logarithm? Oct 13 at 21:40
• You might need to solve a variable first, as thats what it says on there. Oct 13 at 21:41
• @angryavian, thanks, sorry may not have been clear but I need to solve for x, if i subtract 2^x from both sides what happens to the x then? Oct 13 at 21:46
• On the left, the $2^x$ goes away. On the right, since you have $d$ copies of $2^x$ and you are subtracting one, you now have $d - 1$ copies of it. So, on the right you have $2^x(d - 1)$. If you divide both sides by $(d - 1)$, that leaves $2^x$ on the right-hand side. If you take the base 2 logarithm, that will leave just $x$. Oct 13 at 21:51
• @johnny, thanks a lot Oct 13 at 22:03

\begin{align} &ab-c+2^x=2^xd\\ &\Rightarrow{ab-c}=2^{x}\left({d-1}\right)\\ &\Rightarrow{2^{x}}=\left(\frac{ab-c}{d-1}\right)\\&\Rightarrow{x}=\log_2\left(\frac{ab-c}{d-1}\right)\end{align}
The main conditions are: $${d}\neq{1},\quad\left(\frac{ab-c}{d-1}\right)\gt{0}.$$