Apologies if this question has already been answered elsewhere, but I could not find the answer explicitly (this answer appears related - logarithm of a sum or addition)
How would you solve an equation similar to this:
$2^x + 2 = 2^{2x}$
By inspection, the solution is $x=1$, but I am interested in how a solution could be obtained with different values in the equation. For example, how would you solve:
$2^x + 3 = 2^{2x}$ or $2^x + 2 = 2^{3x}$
I already know that a solution can be found using iterative or numeric methods; I am looking for an algebraic or analytic solution to this kind of equation (or an explanation of why none exist).
Thanks in advance!
EDIT:
Ideally a method for solving the above examples would be applicable to a case such as $2^x + 2 = 3^{x}$.
HINT
The standard method is to consider $t=2^x$ then
$$ 2^x + 2 = 2^{2x} \implies t^2-t-2=0$$
then solve for $t$ by quadratic equation and for
$$ 2^x + 2 = 2^{3x} \implies t^3-t-2=0$$
we need to solve first a cubic equation.
Once we have valid solutions for $t>0$ we can obtain $x=\log_2 t$.