How do I solve this exponential equation? $5^{x}-4^{x}=3^{x}-2^{x}$ How do I solve this exponential equation?
$$5^{x}-4^{x}=3^{x}-2^{x}$$
 A: Hint:
$$\frac{5^x-4^x}{5-4} = \frac{3^x-2^x}{3-2}$$

Now use the Lagrange Mean Value Theorem over the intervals $(2,3)$ and $(4,5)$. The equality holds only when $x \in \{0,1\}$.

A: Let $f:[0.25,2] \to \mathbb R \,;\, f(a)=(3.5-a)^x+(3.5+a)^x \,.$
Then $f'(a)=x[(3.5+a)^{x-1}-(3.5-a)^{x-1}]$.
Claim: If $x \notin \{ 0,1 \}$ we have $f'(a) \neq 0 \forall a $.
Indeed, in this case $f'(a) =0 \Rightarrow (3.5+a)^{x-1}=(3.5-a)^{x-1} \Rightarrow a=0$ which is not in our domain.
This proves that for $x \neq 0,1$, $f$ is one to one on our domain, and hence
$$f(0.5) \neq f(1.5) \Rightarrow 3^x+4^x \neq 2^x+5^x $$
A: $$5^x - 4^x = \int_4^5 x y^{x-1} \,dy$$
$$3^x - 2^x = \int_2^3 x y^{x-1} \,dy$$
$$= \int_4^5 x (y-2)^{x-1} \,dy$$
So the difference between $5^x - 4^x$ and $3^x - 2^x$ is
$$ \int_4^5 x (y^{x-1} - (y-2)^{x-1})\,dy$$
In the integrand here, since $y \rightarrow y^{x-1}$ is monotone whenever $x \neq 1$, the
expression $(y^{x-1} - (y-2)^{x-1})$ will either be always negative or always positive if $x \neq 1$, in which case the integral itself will be nonzero unless $x = 0$. 
We conclude that as long 
as $x \neq 0$ or $1$, $(5^x - 4^x) - (3^x - 2^x)$ is nonzero. Hence $3^x - 2^x = 5^x - 4^x$ only when $x = 0$ or $1$.
After writing all this out, I probably prefer the mean value theorem approach, but hey it's good to have more than one way of looking at a problem.
A: Only method I can think of is trying graphically. Since their slopes vary a lot after $(1,1)$ and before $(0,0)$, all possible intersections can lie between these two points.
How to find all those possible intersections is something I'd need to work out.
