Solve $5^{x/2}-2^x=1$

Question

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

Answer 1

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

$$x=2$$

Answer 2

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 $x=2$