How to find the value of an unknown exponent? E.g. I have the question:
$$2^{4x+1} = 128$$
I solved this by knowing that $128 = 2^7$ and therefore $x$ must equal $1.5$. 
However, is there a way of solving this without knowing that $128 = 2^7$? 

... and without logarithms or knowing any powers of $2$ other than the most trivial one ...  \begin{align*}
2^{4x+1} &= 128  \\
2^{4x+1-1} = 2^{4x} &= 128/2 = 64  \\
2^{4x-1} &= 32  \\
2^{4x-2} &= 16  \\
2^{4x-3} &= 8  \\
2^{4x-4} &= 4  \\
2^{4x-5} &= 2  \\
2^{4x-6} &= 1 = 2^0 \text{,}  \\
\end{align*}
so  $4x-6 = 0$ and $x = 6/4 = 3/2$.
