Find all solutions of the equation $3 \cdot 2^{x+2}+5^x=8\cdot 3^x+5$ 
Find all solutions of the equation
  $$3 \cdot 2^{x+2}+5^x=8\cdot 3^x+5$$

My work so far:
Let $f(x)=3 \cdot 2^{x+2}+5^x-8\cdot 3^x-5$
$f'(x)=12\cdot 2^x\ln2+5^x\ln5-8\cdot3^x\ln3$

Your equation can be changed to $$5(5^{x-1}-1)=24(3^{x-1}-2^{x-1})$$
At least $x=0$, $x=1$ and $x=3$ are solutions.
If your question is a Diophantine problem:
With $n_1,n_2\in\mathbb{N}$ you have the conditions $$3^{x-1}-2^{x-1}=5n_1$$ $$5^{x-1}-1=24n_2$$ 
EDIT: 
$x\in\mathbb{R}$ o.k.
But with the values 0,1,3 you can check 
$$5(5^{x-1}-1)<,=,>24(3^{x-1}-2^{x-1})$$ 
with $x<0$, $0<x<1$, $1<x<3$ and $x>3$
