Solve $2^{0.5x}=3^{-0.5x}+\frac{5}{3}$.

Question

thomas5267 · Answer

@Kainui The answer is 2 but I couldn't solve it without using graphic display calculator.

thomas5267 · Answer

@ganeshie8 @ikram002p @Marki

anonymous · Answer

yes its 2

thomas5267 · Answer

But how to solve it without using calculator?

anonymous · Answer

do u have to give an explanation

thomas5267 · Answer

It was a past paper question in a paper which prohibits the use of calculator.

anonymous · Answer

i dont know then really im sorry

mathmath333 · Answer

$\large \color{black}{
2^{0.5x}=3^{-0.5x}+\frac{5}{3} \hspace{.33em}\~\ 
2^{x/2}=3^{-x/2}+\frac{5}{3} \hspace{.33em}\~\ 
2^{x/2}-3^{-x/2}=\frac{5}{3} \hspace{.33em}\~\ 
2^{x/2}-\dfrac{1}{3^{x/2}}=\frac{6-1}{3} \hspace{.33em}\~\ 
2^{x/2}-\dfrac{1}{3^{x/2}}=2-\frac{1}{3} \hspace{.33em}\~\ 
\normalsize \text{from this point it is evident that}\quad  x/2=1\implies x=2\hspace{.33em}\~\ 
}$

ganeshie8 · Answer

Neat :)

thomas5267 · Answer

So there is no general method to solve this kind of equations?  For exmaple, does $2^{0.5x}=3^{-0.5x}+7$ have a nice solution?