I am new to logarithms, I have the following equation

$ \ \sqrt{\log_{10}(-x)} = \log_{10}\sqrt{x^2} \ $

I tried by squaring both sides, which yielded me

$ \ \log_{10}(-x) = ( \log_{10}\vert x \vert )^2 \ $


$ \ \log_{10}(-x) = ( \log_{10}\vert x \vert )^2 \ $

Or $ \ \log_{10}(-x) = ( \log_{10}\vert x \vert ) ( \log_{10}\vert x \vert ) \ $

Or $ \ \log_{10}(-x) = ( \log_{10}2\vert x \vert ) \ $

Which on further solving, doesn't gives me the solutions. What is the correct way?

  • $\begingroup$ What should I do then? I'm not cheating at homework $\endgroup$ – Ravi Prakash Jul 4 '17 at 4:14
  • $\begingroup$ Convince this community why we should help you? What makes your post useful to us? What do we gain by answering it? $\endgroup$ – The Great Duck Jul 4 '17 at 4:16
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    $\begingroup$ @Typhon, that's a bit harsh, don't you think? I've seen many posts with far less work and/or thought slide. He has even shown some work and thoughts. Not every post on M.SE needs to be "useful to us" $\endgroup$ – Ravi Jul 4 '17 at 4:17
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    $\begingroup$ @RaviPrakash do not worry about downvotes or rep; as long as your questions are answered, that is all that matters. More than that, do not worry about asking a lot of questions about a similar topic. Just do your best and show your efforts (as you have done so). $\endgroup$ – TomGrubb Jul 4 '17 at 4:28
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    $\begingroup$ @RaviPrakash you're welcome, but I fail to see where I was being "kind" nor "un-kind". I am only voicing my complaint towards your behavior. Nothing more, nothing less. $\endgroup$ – The Great Duck Jul 4 '17 at 4:37

It seems like you mixed up your log rules. Here's one way to go about it: you know $log_{10}$ of a negative number is undefined, so $x$ must be negative for $\log_{10}(-x)$ to make sense. Further, you know $\sqrt{x^2}=|x|$; thus the question turns into solving (by setting $y=-x$) $$ \sqrt{\log_{10}(y)}=\log_{10}(y) $$ Now when do we have that the square root of a number must be equal to that number? We get $\log_{10}(y)=0$ or $\log_{10}(y)=1$. Can you go from here?

  • $\begingroup$ $\log_{10}(x)$ is very well defined for negative x's. Please explain where that assumption is ever justified? $\endgroup$ – The Great Duck Jul 4 '17 at 4:20
  • $\begingroup$ Looks like a case of telepathy. Maybe I should have checked your answer while I was finishing mine off. $\endgroup$ – Lubin Jul 4 '17 at 4:21
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    $\begingroup$ @Typhon The OP says they are "new to logarithms;" thus I am assuming that they have not heard of branch cuts and complex logs. $\endgroup$ – TomGrubb Jul 4 '17 at 4:22
  • $\begingroup$ @Lubin hopefully this is a case of great minds thinking alike and not fools never differing :) $\endgroup$ – TomGrubb Jul 4 '17 at 4:23
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    $\begingroup$ @Typhon, logarithms are definable for negative numbers, but they are not defined. That is, deciding what $\log(-1)$ shall be requires a choice, on which honest people of good will may easily disagree. $\endgroup$ – Lubin Jul 4 '17 at 4:23

First, you must remember that $\sqrt{x^2\,}=|x|$. Second, you must remember that logarithms are defined only for positive inputs, so on the left hand side, we need $x$ to be negative, which means that $|x|=-x$. As a result, your equation to solve becomes $\sqrt{\log_{10}(-x)}=\log_{10}(-x)$. Calling $z=\log_{10}(-x)$, the equation becomes $\sqrt z=z$, which happens only for $z=0,1$, as you can easily check. Thus, in case $z=0$, you have $-x=1$ and $x=-1$, while in case $z=1$, you have $-x=10$ and $x=-10$.

  • $\begingroup$ logarithms are defined for negative inputs. $\endgroup$ – The Great Duck Jul 4 '17 at 4:21

First of all, $\log a \log b \neq \log ab$ in general.

$$\log (-x) = (\log |x|)^2$$

From LHS, $-x > 0 \implies x < 0$.

$$\log(-x) = (\log(-x))^2 \implies \log(-x) = 0 \text{ or } 1 \implies x = -1 \text{ or }-10$$

Edit: If the domain of $x$ is not restricted to real numbers then let $x = r e^{i\theta} \implies -x = re^{i(\theta+\pi)}$,

$$\log(re^{i(\theta+\pi)}) = (\log r)^2 \implies \log r + i(\theta+\pi) = (\log r)^2$$

$$\implies \theta = -\pi \wedge \log r = (\log r)^2 \implies \theta = -\pi \wedge r = 1 \text{ or } 10$$

  • $\begingroup$ What prevents -x from being negative? $\endgroup$ – The Great Duck Jul 4 '17 at 4:17
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    $\begingroup$ Domain of $\log x$ is $x > 0$. $\endgroup$ – Dhruv Kohli - expiTTp1z0 Jul 4 '17 at 4:18
  • $\begingroup$ That is blatantly false. log(-1) is very well defined. $\endgroup$ – The Great Duck Jul 4 '17 at 4:21
  • $\begingroup$ I think the OP needs a solution for $x \in \mathbb{R}$ $\endgroup$ – Dhruv Kohli - expiTTp1z0 Jul 4 '17 at 4:23
  • $\begingroup$ I wouldn't make that assumption though without them saying so. I've seen stranger. Plus, are you sure allowing for negative x's forces a solution to be off of the real line? $\endgroup$ – The Great Duck Jul 4 '17 at 4:24

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