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How do I solve this exponential equation?

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$x=1{}{}{}{}{}$ –  Will Jagy Feb 2 '13 at 4:08
$x=0,1$ seems to be the only real solution. for $x\gt 1$, the L.H.S. growth rate is more than that of R.H.S. and for $x\lt 0$, L.H.S. $\lt $ R.H.S –  Aang Feb 2 '13 at 4:09
@WillJagy You forgot $x = 0$ –  hjpotter92 Feb 2 '13 at 4:09
No, it is a religious prohibition. Well, it will be, once I have finished designing the religion. –  Will Jagy Feb 2 '13 at 4:10
@BackinaFlash: Will forgot nothing. :-) –  Brian M. Scott Feb 2 '13 at 4:11

4 Answers 4

up vote 29 down vote accepted

Hint:: $\displaystyle \frac{5^x-4^x}{5-4} = \frac{3^x-2^x}{3-2}$

Now Use LMVT (Lagrange mean value theorem) in $(2,3)$ and $(4,5)$

These two are equal only when $x = 0$ and $x=1$

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$$5^x - 4^x = \int_4^5 x y^{x-1} \,dy$$ $$3^x - 2^x = \int_2^3 x y^{x-1} \,dy$$ $$= \int_4^5 x (y-2)^{x-1} \,dy$$ 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.

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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.

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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

$$f(0.5) \neq f(1.5) \Rightarrow 3^x+4^x \neq 2^x+5^x $$

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