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 $3$.
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Can you write what you've done? – mathlove Jun 29 '15 at 18:50
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I will post a picture : – zivce Jun 29 '15 at 18:52
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Pic – zivce Jun 29 '15 at 18:54
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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 $x=100$.
Plutoro
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I guess it is something like this $$-log^2_{10}(x)$$. I don't know how to solve this there is a picture above this post, I tried that. – zivce Jun 29 '15 at 19:05
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I got here but what should I do next. I need to get the sum of solutions by $$x$$ – zivce Jun 29 '15 at 19:19
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1I found the mistake we both made. We said $\log(9)=3\log(3)$, not $2\log(3)$. See my edited answer. – Plutoro Jun 29 '15 at 19:31
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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.
Theo Bendit
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You just want the sum of solutions, right? You can get that by using Vieta's formulae. It will be $3 + \log_{10}(3)$. – Theo Bendit Jun 29 '15 at 19:27