Find all real number satisfying $10^x+11^x+12^x = 13^x+14^x$ Find all real number $x$ satisfying $10^x+11^x+12^x = 13^x+14^x$    


My Work
Dividing by $13^x$ we get 
$$\left( \frac{10}{13} \right)^x + 
\left( \frac{11}{13} \right)^x + 
\left( \frac{12}{13} \right)^x
= 1 + \left( \frac{14}{13} \right)^x$$
The LHS is a decreasing function of $x$ and the RHS is an increasing function of $x$. So there is only one intersection in their graph. I am looking for a formal way to find the root. I know that $x=2$ works. But how to formally find this root?
As you have done, $$\left( \frac{10}{13} \right)^x + 
\left( \frac{11}{13} \right)^x + 
\left( \frac{12}{13} \right)^x
= 1 + \left( \frac{14}{13} \right)^x$$ 
has a decreasing function on the left and an increasing function on the right. So there is only one solution. 

If you want to find them, just plug in some values. Plug in $x=1,2, \dots$. Now, we see $x=2$ works. $$10^2+11^2+12^2=13^2+14^2=365$$
Problems like this usually have a solution over the integers, or don't have a closed form solution at all. The OP pointed out that one might make a mistake if doing it by hand. However, plugging in certain values is helpful in most cases. This is true since it would help approximate the solution using IVT. 

I doubt that there is a truly formal/analytical way to solve this equation. If there was, we could express the roots of $$2^x+3^x+4^x=5^x+6^x$$
using some well known functions. However, this kind of equation is known to have no closed form solution. We just know by plugging the values that $x$ lies somewhere between $0$ and $1$. So this is impossible. 

However, you could approximate the roots using Newton's Method or a similar method.
I just solved my problem. A well known fact that 

$$a^2 +(a+1)^2 + \cdots + (a+k)^2 = (a+k+1)^2 +(a+k+2)^2+ \cdots (a+2k)^2$$
Satisfies when $a = k(2k+1)$.
So in my question $a = 2(2\cdot2+1)$, So without any thinking $x=2$ is an answer.
This helps suppose the equation was - 
$$55^x+56^x+57^x+58^x+59^x+60^x=61^x+62^x+63^x+64^x+65^x$$
Then it is hard to just substitute $x = 1,2,3,..$ and check? But using this fact without any thinking we can say $x=2$, as $55 = 5(5\cdot2+1)$.