Sum of solutions of this exponential equations How to solve this : 
$$x^{3-\log_{10}(x/3)}=900$$
I tried log on both sides and got nothing with exponent of $x$ and $3$. 

Go ahead and take the $\log_{10}$ on both sides:
$$3\log_{10}(x)-\log_{10}(x)^2+\log_{10}(x)\log_{10}(3)=\log_{10}(900).$$
Now solve the quadratic.
Let $y=\log_{10}(x).$ Then this quadratic is
$$y^2-(\log_{10}(3)+3)y+\log_{10}(900)=0.$$
Applying the quadratic formula, we get
$$y=\frac{3+\log_{10}(3)\pm\sqrt{\log_{10}(3)^2+6\log_{10}(3)+9-4\log_{10}(900)}}{2}$$
$$=\frac{3+\log_{10}(3)\pm\sqrt{\log_{10}(3)^2+6\log_{10}(3)+9-8\log_{10}(3)-8}}{2}$$
$$=\frac{3+\log_{10}(3)\pm\sqrt{\log_{10}(3)^2-2\log_{10}(3)+1}}{2}$$
$$=\frac{3+\log_{10}(3)\pm(\log_{10}(3)-1)}{2}$$
$$=\cases{1+\log_{10}(3)=\log_{10}(30)\\2}$$
Thus, $x=30$ or $x=100$.
