Solve for $x$: $10^x=x+100$

Question

I don't know how to solve it: $$10^x=x+100$$ The original question was how many answers are there, so I drew the functions, then I see there are 2 answers (cuts?). I really want to know how to solve and know the actual answers, but I don't really know how to solve it. In general how do I solve equations which got $x$, $a^x$, $c$, and more...

*Which tags should I put? *How do I write in the math font? I always see the math font, but I don't know how to do it.

Answer 1

Use an iterative method like

$$x=\log_{10}(x+100).$$

Starting from any initial guess $x>-100+1.0000000001\times10^{-100}$ and you'll get the answer in a few iterations: $x=2.008636944$. Another answer is very close to $x=-100$, which can be reached by the iteration

$$x=-100+10^x$$

starting from $x=-100$ as the initial guess. Then the solution is $x\approx-100+10^{-100}$.

Answer 2

I can't believe that WA didn't succeed.

$$10^x=x+100$$

$$10^{-100}e^{\log10(x+100)}=x+100$$

$$-10^{-100}\log10=-\log10(x+100)e^{-\log10(x+100)}$$

$$W(-10^{-100}\log10)=-\log10(x+100).$$