I want to mention that it is not my homework, just want to solve for fun. I appreciate any hint how to solve it. The exponential equation is given:
$2^x + 3^x = 10000$
My initial thought was to use such transformation: $2^x + 2^{\log_2{3^x}} = 10000$, but it gives nothing for me
If $(2, x+1)=1$
also $(3, x+1)=1$ we may write:
$2^x + 3^x ≡ 2 mod (x+1)$ or $2^x+3^x= y(x+1) +2=10000 $
⇒ $y(x+1)=9998$
Now we have a system of equation:
$y(x+1)=9998$
$2^x+3^x=10000$
That is the solution is the coordinates of intersection of two curves; one a hyperbola, another an exponent curve, hence the equation can be solved by sketching.
Also:
$2^8+3^8=6817$
$2^9+3^9=20195$
⇒ $8 < x < 9$
by try and error we can find that $x≈ 8.3535$
$2^{8.3535}+3^{8.3535}≈ 10001.66$
Hence $y=\frac{9998}{9.3535}≈1069$