3^x+x = √3

3^x+x = √3

Solution with iteration: $$3^x = \sqrt{3} - x \quad | \quad \ln{()} \\\\
x\ln{(3)} = \ln{ ( \sqrt{3} - x ) } \\\\
\textcolor[rgb]{1,0,0}{\boxed{x_{new} = \dfrac{ \ln{ ( \sqrt{3} - x_{old} ) } }{\ln{(3)} } } }\\\\
x_{old} \small{\text{ starts with 1} } }$$

$$\begin{array}{r|l|r}
\hline
n & x_{new} & x_{old}\\
\hline
1 & -0.28390849201 & 1 \\
2 & 0.63816431742 & -0.28390849201 \\
3 & 0.08168208459 & 0.63816431742 \\
4 & 0.45602869783 &0.08168208459 \\
5 & 0.22186854696 &0.45602869783 \\
6 & 0.37522823128 &0.22186854696 \\
7 & 0.27775551797 & 0.37522823128 \\
8 & 0.34090411159 & 0.27775551797 \\
9 & 0.30049579054 & 0.34090411159 \\
10 & 0.32655858818 &0.30049579054 \\
11 & 0.30983412349 & 0.32655858818 \\
12 & 0.32060145623 & 0.30983412349 \\
13 & 0.31368398912 & 0.32060145623 \\
14 & 0.31813414565 &0.31368398912 \\
15 & 0.31527376081 & 0.31813414565 \\
16 & 0.31711333488 & 0.31527376081 \\
17 & 0.31593069256 & 0.31711333488 \\
18 & 0.31669117690 & 0.31593069256 \\
19 & 0.31620222924 & 0.31669117690 \\
20 & 0.31651662459 & 0.31651662459 \\
21 & 0.31631447956 & 0.31651662459 \\
22 & 0.31644445678 & 0.31631447956 \\
23 & 0.31636088486 & 0.31644445678 \\
24 & 0.31641462028 & 0.31636088486 \\
25 & 0.31638006963 & 0.31641462028 \\
26 & 0.31640228506 & 0.31638006963 \\
27 & 0.31638800101 & 0.31640228506 \\
28 & 0.31639718538 & 0.31638800101 \\
29 & 0.31639128001 & 0.31639718538 \\
30 & 0.31639507705 & 0.31639128001 \\
31 & 0.31639263563 & 0.31639507705 \\
32 & 0.31639420541 & 0.31639263563 \\
33 & 0.31639319608 &0.31639420541 \\
34 & 0.31639384506 &0.31639319608 \\
35 & 0.31639342778 & 0.31639384506 \\
36 & 0.31639369608 & 0.31639342778 \\
37 & 0.31639352357 & 0.31639369608 \\
38 & 0.31639363449 & 0.31639352357 \\
39 & 0.31639356317 & 0.31639363449 \\
40 & 0.31639360903 & 0.31639356317 \\
41 & 0.31639357954 & 0.31639360903 \\
42 & 0.31639359850 & 0.31639357954 \\
43 & 0.31639358631 & 0.31639359850 \\
44 & 0.31639359415 & 0.31639358631 \\
45 & 0.31639358911 & 0.31639359415 \\
46 & 0.31639359235 & 0.31639359235 \\
47 & 0.31639359026 & 0.31639359235 \\
48 & 0.31639359160 & 0.31639359026 \\
49 & 0.31639359074 & 0.31639359160 \\
50 & 0.31639359130 & 0.31639359130 \\
51 & 0.31639359094 & 0.31639359130 \\
52 & 0.31639359117 & 0.31639359094 \\
53 & 0.31639359102 & 0.31639359117 \\
54 & 0.31639359112 & 0.31639359102 \\
55 & 0.31639359106 & 0.31639359112 \\
56 & 0.31639359109 & 0.31639359106 \\
57 & 0.31639359107 & 0.31639359109 \\
58 & 0.31639359109 & 0.31639359107 \\
59 & 0.31639359108 & 0.31639359109 \\
60 & 0.31639359108 & 0.31639359108 \\
... & ... & ... \\
\hline\end{array}$$

$$\boxed{x= 0.31639359108 }$$

$$\\ f(x) = \frac{ \ln \left( \sqrt{3} - x \right) } {\ln{(3)}} \\
g(x) = x \\ \\
\small{\text{ We set }} f(x) = g(x) \small{\text{ and iterate the cut between line and our function. }}$$

The convergence is given if the derivation $$\boxed{|f'(x_{Start})| < 1}$$

$$\\ f(x) = \frac{ \ln \left( \sqrt{3} - x \right) } {\ln{(3)}} \\\\
f'(x) = \frac{1}{\ln{(3)}} * \frac{(-1)}{ \left( \sqrt{3} - x \right) }$$

The graphical soln shows the real solution to be approx x=0.316

Here is the graph

https://www.desmos.com/calculator/msrf4kimfo

BUT I do not know how to do this algebraically.   

Thanks Heureka.

That makes sense :)