Here is the question:Solve $5^{\frac{x}{2}}-2^x=1$
How i tried:I was just looking at the equation and was trying different values of x and got x=2 .But the way to reach answer was not promising so I decided to graph it and observed that the function is ever increasing from (-$\infty,\infty$) ,so the graph cuts $y=1$ only once at $x=2$.This was my way to solve the question but is there some other algebric way to solve it?
The function $f(x)=5^{x/2}-2^x$ isn't increasing (It is decreasing to the left of Dr. SG's critical point). But it is less than $0$ when $x<0$.
So we have that $5^{x/2}-2^x<0$ when $x<0$ (so it can't equal $1$), and $5^{x/2}-2^x$ is increasing when $x>0$, with $f(0)=0$ and $\lim_{x\rightarrow\infty} f(x)=\infty$ so $5^{x/2}-2^x=1$ has exactly one solution.
I don't think such an equation can be solved algebraically in general, but luckily 2 works.