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 :)