Number of solutions

[proven frigate]

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[paper hound]

This can't be solved analytically.
Though, as f''(x) = 2 + (ln(2))^2 2^x is always positive, this means that the function is everywhere convex down.
Moreover, as f(-2) > 0, f(-1) < 0 and f(0) > 0, this means there are two roots: one between -2 and -1 and another between -1 and 0.

You can find them with the numerical method of your choice.

[reef zephyr]

it has no real solutions

[paper hound]

What? No, it has two.

[fickle badger]

@eager fiber has given 1 rep to @paper hound