Exponential function 1st degree function

Question

The question is in Portuguese, I'll try to translate it. 15) How many real roots does this equation have: $2^{x}=x+4$

a) infinitely many

b) one

c) two

d) three

e) four

The answer is letter c) But I couldn't figure out why, and also which are the real roots. I only know that they can be seen as the intersections of the graph of $f(x)=2^{x}$ and $g(x)=x+4$

Answer 1

Let $f(x)=2^x$ and $g(x)=x+4$.

Thus, $f$ is a convex function and $g$ is a linear function, which says that our equation has two roots maximum.

But $f(0)<g(0)$, which says that $f-g$ has two roots.

One of them on $(-4,0)$ and the second on $(0,3)$.

Answer 2

This isn't the intended solution, but the roots can be found in "closed form" using the Lambert W function:

$$ x = \frac{-W(-\ln(2)/16)}{\ln(2)} - 4 $$

Since $-1/e < -\ln(2)/16 < 0$, for real roots you can take either the $-1$ or the $0$ branch of $W$, thus there are two real roots.