How to solve exponential function inequality?

Question

How do I solve the exponential equations like $2^\frac{x}{8}<x$? I can solve this by plotting into graph. But is there any way to do it mathematically? or like $2^x < 100x^2$ . I am trying to actually solve the thing. ie for which set of values the inequality holds...

Answer 1

Let's examine this WolframAlpha output:

From there, it's easy to see geometrically what the solution set looks like. (It sounds like you knew this much.) We also see the solution set expressed in terms of a mysterious function $W$, which is a special function called the Lambert $W$ function. You can click the approximate form button to find that (approximately) $1.1<x<44.559$ but I don't believe that the solution set can be expressed in a simpler form.

Answer 2

Your equations can't be solved algebraically (without involving Lambert's $W$), but numerical instead.

Answer 3

For the first question, the problem is basically to solve the equation $$f(x)=2^{x/8}- x=0$$ The solutions are obtained using Lambert function and the solution are $$x_1=-\frac{8 W\left(-\frac{\log (2)}{8}\right)}{\log (2)}$$ $$x_2=-\frac{8 W_{-1}\left(-\frac{\log (2)}{8}\right)}{\log (2)}$$ which are respectively equal to $1.10$ and $43.56$. So, the inequality is satisfied between these two roots.

In practice, any equation of the form $$a+b x+c \log (d+e x) =0$$ has solutions expressed using Lambert function.

The same happens with the second equation you gave; the three roots of the equations are also given in terms of Lambert function; they correspond to $x_1=-0.10$, $x_2=0.10$ and $x_3=14.32$.