Solving $x + 3^x < 4$ analytically 
I am solving the problems in Michael Spivak's Calculus book. In the Prologue chapter, there is the following problem:
$$x + 3^x < 4$$
I can solve this using graphical means, but not analytically. Is it possible to do the latter?

I can arrive at the solution as follows:
$$3^x < 4 - x$$
$$x < \log_3 (4-x)$$
I then draw the graphs of the functions $\log_3 (4-x)$ and $x$, and I can determine that $x$ needs to be smaller than 1 to be smaller than the $\log_3 (4-x)$.


How do I solve this purely analytically?


As @IV_ writes (in solution 3), standard application of Lambert's W function gives the solution:
$$x = \frac{4 \log (3)-W(81 \log (3))}{\log (3)}$$
Note that $W(81 \log (3)) = \log (27)$, so the full solution gives $x=1$, which can be easily confirmed.
