Sum of solutions of this exponential equations

Question

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 $3$.

Answer 1

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 $x=100$.

Answer 2

Take the log base $10$ of both sides, and let $u = \log_{10}(x)$. Then the expression turns into a quadratic in terms of $u$, which you can solve.