# Is it possible to solve this equation?

Here is an equation that I found is quite impossible to solve without graphing or approximating the answer. $$\sqrt{x} = 1+\ln(5+x)$$ I tried squaring both sides and factoring the $ln$ out, but it was no use. I also tried to get the procedure on WolframAlpha, but it doesn't have a procedure available for me.

-
Pretty sure approximations are required. – zibadawa timmy Nov 7 '13 at 3:44
@zibadawatimmy - So you mean there's absolutely no way to get the exact answer algebraically? – Derek 朕會功夫 Nov 7 '13 at 3:54
Absolutely? I'm not sure I can go that far, simply because I cannot think of a proof of such a statement. However, the basic problem is that the system is using a transcendental function like $\ln$ in a complicated fashion, which in some sense means the equation is inherently "non-algebraic". – zibadawa timmy Nov 7 '13 at 4:00
@zibadawatimmy very nicely said. – Betty Mock Nov 7 '13 at 6:16
It is clear neither $x = 1$ nor $x = -5$ is a solution of $$\sqrt{x} = 1 + \log(5+x) \quad\iff\quad 1\cdot e^{\sqrt{x}-1} - (5+x) \cdot e^0 = 0$$ For other algebraic $x$ differs from $1$ and $-5$, $\sqrt{x}-1$ and $0$ are distinct algebraic numbers and $5+x$ are non-zero algebraic number. By Lindemann-Weierstrass Theorem, the expression in LHS of $2^{nd}$ expression $\ne 0$. i.e. The equation at hand doesn't have any algebraic solution at all. – achille hui Nov 7 '13 at 6:19

As far as I know, there cannot be any solution to this equation because of the simultaneous presence of $\sqrt{x}$ and $x$. The only solution can be obtained using a numerical method such as Newton.
$$x_{new} = x_{old} - \frac{f(x_{old})}{f'(x_{old})}$$
Starting at $x=1$, which we know is very far from the solution (see the graph of the function), the successive iterates are $6.357, 14.608, 16.536, 16.579$. Starting at $x=10$, the successive iterates are $15.968, 16.575, 16.579.$
If the equation were $x^k=1+\log(5+x^k)$, there is one or more analytical solutions in terms of Lambert function.