exponential equation with different bases We have $3^x-5^\frac{x}{2}=4$  My question is what we can do here ? Can we solved it algebraically or we need to notice that $x=2$ and then show that for $x \neq 2$ there aren't any other solutions? 

Your second approach, viz. showing there are no other solutions would also require some algebra.  With $2t = x$, you can write the equation as
$$(4+5)^t = 4 + 5^t$$
This is obvious for $t=1$ i.e $x=2$.
For $t > 1$, we have  $(4+5)^t > 4^t+5^t > 4+5^t$
for $0 < t < 1$, let $y = \frac1t > 1$, then $(4+5^t)^y > 4^y+5 > 4+5 = \left((4+5)^t\right)^y$
and for $t \le 0$, $(4+5)^t \le 1 < 4 < 4+5^t $ 
