I tried to think of it a lot but didn't get any breakthrough. I was trying with substitution method, but things were not fitting in.
Actual answer is 2
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Let $x=2u$, then the equation becomes: $$5^u-2^{2u}=1\Rightarrow 5^u=4^u+1^u$$ Divide both sides by $5^u$, $$1=\left(\dfrac{4}{5}\right)^u+\left(\dfrac{1}{5}\right)^u$$ Easily observe that $u=1$ is one of the answer. However, as $\left(\dfrac{4}{5}\right)^u+\left(\dfrac{1}{5}\right)^u$ is strictly decreasing, so there must only have at most one answer. Therefore the answer is $u=1$, which means that $x=2$
MafPrivate
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1if you see first answer is more simple – user3290550 Oct 11 '19 at 23:30
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I want to show that there is only one answer @user3290550 – MafPrivate Oct 11 '19 at 23:30
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So I write a bit long, and I hope you will know that $x=2$ is the only answer. @user3290550 – MafPrivate Oct 11 '19 at 23:32
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@user3290550 How would you prove that 'since they all have the same exponential, the exponential must be 1' with a simple answer? – Toby Mak Oct 12 '19 at 00:17
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Rewrite,
$$5^{x/2}-2^x=1$$
as
$$5^{x/2}=4^{x/2} + 1^{x/2}$$
Since $5=4+1$ and they all have the same exponential, the exponential must be 1,
$$\frac x2=1$$
which yields
$$x=2$$
Quanto
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can you explain"since they all have the same exponential, the exponential must be 1" – MafPrivate Oct 12 '19 at 04:17
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@IsaacYIUMathStudio What he's done is to simply explain the observation that one solution is $x/2=1.$ The only thing is that he didn't explain why this is the complete solution. – Allawonder Oct 12 '19 at 06:01
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Issac, may be a little pretentious providing a similar answer from which you knew the answer to your comment. – Quanto Oct 12 '19 at 13:52