Solving an exponential equation without the quadratic formula High school math student here. In my homework I was asked to solve $16^x +4^{x+1} - 3= 0$ and I used substitution to get $x=\log_4{(-2+\sqrt7)}$. However, this was in the chapter on logarithms. How can I solve this without recourse to the quadratic formula etc, that is, using only logarithms and their properties and exponential functions and their properties? If I can't, how would one prove this is impossible?

(Hopefully someone still finds this useful three and a half years later!)

This is the best way I can think to do this problem without the quadratic formula, and I suspect it may be similar to the way in which you did it.

$16^x+4^{x+1}-3=0$ can be rewritten as $(4^x)^2+4(4^x)-3=0$, and we can make the substitution $u=4^x$ to arrive at $u^2+4u-3=0$. Unfortunately, that doesn't factor nicely, so the next best option (assuming we're not allowed to use the quadratic formula) is to complete the square, as Rory Daulton suggested in a comment. $$\begin{align}u^2+4u-3&=u^2+4u+4-7\\\&=(u+2)^2-7\\\\(u+2)^2-7&=0\\\\(u+2)^2&=7\\\u+2&=\pm\sqrt{7}\\\u&=-2\pm\sqrt{7}\end{align}$$

Now we substitute back in: $$\begin{align}4^x&=-2\pm\sqrt{7}\\\x&=\log_4(-2+\sqrt{7})\end{align}$$