# Solve some unusual log/exponential equations

I understand about log and exponential equations/functions, but I can't solve these (the numbers are just examples, of course):

$4^x = x + 10$
$x^x = 3$
$(2x + 3x^2)^{x + 1} = (x - x^3)^{x^2}$

Are there specific algebraic steps that I can follow so that I solve them? Or, if not, what are the other means to find $x$ in each?

• do you know about the Lambert W function? – Deven Ware Nov 27 '13 at 23:55
• I didn't, but I am looking it up right now. It seems pretty interesting, and way more complex than I imagined... – Luan Nico Nov 28 '13 at 0:09
• Are you looking for analytical solutions (which involve, as told, Lambert function) or for numerical solutions (which would involve Newton procedures) ? If the latest is of any interest tou you, let me know and I shall develop. – Claude Leibovici Nov 28 '13 at 4:59

$$4^x=x+10\quad|\cdot(-1)\cdot4^{-(x+10)}\iff-4^{-10}=-(x+10)\cdot4^{-(x+10)}\quad|\cdot\ln4\iff$$
$$-4^{-10}\ln4=-(x+10)\ln4\cdot e^{-(x+10)\ln4}\iff-(x+10)\ln4=W(-4^{-10}\ln4)\iff$$
$$x=-\frac{W(-4^{-10}\ln4)}{\ln4}-10.$$