Solving exponential equation

Question

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?

I have also found x=2 ,but what was your way to found x=2 @Dr.SonnhardGraubner — Kartik Watwani, Jul 18 '15 at 09:58
Rewriting it like this, it is easy to see the solution $x = 2$: $$ 1 = 5^{x/2} - 2^x =5^{x/2} - 4^{x/2} $$ — izœc, Jul 18 '15 at 10:01
@izœc Ya seeing the solution is not the problem,the problem is to find it using algebra. — Kartik Watwani, Jul 18 '15 at 10:04
@Dr.SonnhardGraubner Ya i have already differentiated the function but it doesn't give any promising results — Kartik Watwani, Jul 18 '15 at 10:05
@KartikWatwani. izoec's answer uses algebra and, to me, it is the best approach and solution for the question. — Claude Leibovici, Jul 18 '15 at 13:03
This can be rewritten as $$a^x-1=b^x$$ with $a=\frac12\sqrt5$ and $b=\frac12$. Note that $a>1$ hence $x\mapsto a^x-1$ is increasing. Note that $b<1$ hence $x\mapsto b^x$ is decreasing. Using the value at $x=0$ and when $x\to\infty$, one sees there exists a unique root $x$, which is positive. Finally, identifying $x=2$ is already done multiple times on this page. — Did, Jul 19 '15 at 08:55

score 2 · Answer 1 · answered Jul 18 '15 at 10:40

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.

Solving exponential equation

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