Solving $x + 3^x < 4$ analytically

Question

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?

Answer 1

$x+3^{x}$ is a continuous strictly increasing function. Hence $\{x: x+3^{x}<4\}=(-\infty, t)$ where $t$ is the unique real number $t$ such that $t+3^{t}=4$. Note that $t=1$!

Answer 2

$$x+3^x-4=0$$

1.) Solution as series

To solve the equation analytically as a series, you can use Lagrange inversion.

2.) No solution with elementary inverses

The equation is a zeroing equation of an elementary function.

Powers with irrational exponents are transcendental functions. Therefore the equation isn't an algebraic equation, it's a transcendental equation:

$$x+e^{\ln(3)x}-4=0$$

Because $x+e^{\ln(3)x}-4$ is a polynomial of two algebraically independent monomials simultaneously ($x,e^{\ln(3)x}$), the function $\mathbb{R}\to\mathbb{R}, x\mapsto x+e^{\ln(3)x}-4$ seems to have no elementary partial inverse. For the same reason, we cannot rearrange the equation by applying only finite numbers of only elementary functions (elementary operations) which we can read from the equation.
But we can see if we can guess solutions.

3.) Solution in closed form with Lambert W

The equation is solvable by applying Lambert W. By some simple rearrangings, we can bring the equation into a form that is solvable by Lambert W.
The only real solution is: $$x=\frac{4\ln(3)-W(81\ln(3))}{\ln(3)}.$$

That's equal to $1$, but I don't know the methods to show this.

Answer 3

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.