A counter-example (based on my misunderstanding) is, say, if $(-3)^3=-27$ then why is it not the case that $log_{-3}(-27)=3$?
Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$), are true, why aren't logarithms defined for negative numbers?  Why can't $b$ be negative? Take $(-2)^3 = -8$ for example. Turning that into a logarithm, we get $3 = \log_{(-2)}{(-8)}$ which is an invalid equation as $b$ and $x$ are not positive! Why is this?
Can Cycles physically mix colours? Does an unpublished document you upload to ResearchGate count as prior publication when submitting to a journal? On ResearchGate, if you upload something like a thesis, and then it says publish resources, does this mean the thesis becomes properly published in the sense of the usual requirements of journals that there be no prior publication of the submitted work?   I ask because I’m drafting thesis chapters into journal articles and did not choose to publish the thesis as a whole work in order to do so, so I want to make sure that by clicking publish resources I am not actually publishing the copy I’ve uploaded, I’m just making it publicly available? Does this mean it becomes published despite it not technically being a published work? (If so, I’ll just remove the copy I uploaded). For example, you're a fashion designer who wants to get the opinion of both a professional and someone who doesn't know much about fashion. What's a term for the latter person? Why must a singularity form inside a black hole? Orthonormal basis with specific norm I need to show that $v_1,...,v_n$ is basis for $V$ whenever $e_1,...,e_n$ is an orthonormal basis for V and $v_1,...,v_n$ are vectors in $V$ such that $$\left\Vert e_i-v_i\right\Vert &lt; \frac{1}{\sqrt{n}}.$$ I know already that I only need to prove that they are lineary independent. Also I tried to prove it by contradiction, but I got stuck fairly fast. Any help, or hint how to prove it (with ot without contradiction) will be appriciated. Are all multiplicative functions additive? Are intermediate rings of finitely generated ring extensions also finitely generated? Display historical characters "uͤ", "oͤ" and "aͤ"? Add chat rooms for Android app Being able to join chat rooms from the Android app would be a plus! It would truly make us connected on the go, instead of having to sit down at our desks and open a browser window. This will allow more users to fully utilize the chat system and join events no matter where they are. A number of times I have encountered in text-books and articles that neutrinos might contribute only a small fraction to dark matter. The reason has to do with the fact that if all of the dark matter consisted of neutrinos, then small-scale structures in the Universe could not have formed yet, because, as they say, neutrinos "wash out" small fluctuations. However, none of these texts provided a reference to any specific sources explaining in detail what is meant by "washing out". After all, neutrinos are notorious in their weak interaction with baryonic matter, so if there is a small-scale fluctuation of baryons, then how background neutrinos can prevent it from growing further if they practically do not interact with baryons? I guess the question boils down to calculating cross-sections of interactions at specific temperatures. I would appreciate comments and references to sources addressing this particular issue. Understanding ROC curve It's a bit difficult to implement a deep object copy function. What steps you take to ensure the original object and the cloned one share no reference? Do you think that "was going to be" or "would be" could be used as past tense of will?     I thought this book was going to be your choice.        I thought this book would be your choice.   Do these two sentences basically mean the same thing or are they slightly different? Add a confirmation before moving comments to a chat Why apt can't find the source code of any package? When certain nationalities of people want to travel in the Schengen zone, they are required to apply for a Schengen visa. This is a straight-forward process for most, but when an applicant has had problems with removals or refusals they may be reluctant to apply because they want to avoid the complexities of serial visa refusals. This reluctance may lead some into wondering how their history can be expunged, and this line of thought inevitably leads to the idea of getting their passport replaced with a new one.  This sounds like a great formula for people who have had prior refusals or removals.  In essence they can become 'reincarnated' as a first-time traveller and hence avoid the unpleasantness of revealing their prior interactions with the Schengen authorities.  The relevant parts of the application form are shown here...    Nothing asks about a prior removal or refusal, and the applicant is left with simple denials for 3 questions in order to present themselves as a first-time traveller.  Assuming the person is happy to set aside the moral issues associated with denying their history, are there any holes to this strategy?  This question is scoped to Schengen visa applications so that it will not be unmanageably broad, but answers from other regimes (particularly the affluent Commonwealth and the USA) are acceptable as long as the main points are relatively consistent with Schengen. Examples and further results about the order of the product of two elements in a group Let $G$ be a group and let $a,b$ be two elements of $G$. What can we say about the order of their product $ab$?   says "not much":     There is no general formula relating the order of a product $ab$ to the orders of $a$ and $b$. In fact, it is possible that both $a$ and $b$ have finite order while $ab$ has infinite order, or that both $a$ and $b$ have infinite order while $ab$ has finite order.   On the other hand no examples are provided. $(\mathbb{Z},+), 1$ and $-1$ give an example of elements of infinite order with product of finite order. I can't think of any example of the other kind! So:   What's an example of a group $G$ and two elements $a,b$ both of finite order such that their product has infinite order?     Wikipedia then states:     If $ab = ba$, we can at least say that $\mathrm{ord}(ab)$ divides $\mathrm{lcm}(\mathrm{ord}(a), \mathrm{ord}(b))$   which is easy to prove, but not very effective. So:   What are some similar results about the order of a product, perhaps with some additional hypotheses? Who in the DC Universe can stop Superman if he has a breakdown? Is there anything or anyone in the DC universe that can defeat Superman just in case he goes berserk on everybody?  Maybe Batman stores kryptonite in his Cave or something. I don't know if Batman actually has kyptonite, the example was just a theoretical example.  Who can stop Superman if needed, and how? Let $f_1 : [-1, 1] \rightarrow \mathbb{R};\: f_1(0) = 0 $ be a continuously differentiable function and $\lambda &gt; 1$. Consider the sequence of functions defined inductively by $f_k(x) := \lambda f_{k-1}\left(\dfrac{x}{\lambda}\right);\: k \geq 2;\: x\in [-1, 1]$. Find the pointwise limit of the sequence of functions $(f_n)$.   I figured out that $f_k(0)=0 \:\forall\: k\implies f_k\rightarrow 0 $ when $ x=0$. Also $f_k$ is continuously differentiable $\forall \:k$.   $\Bigg($Since $\quad\displaystyle\lim_{h\rightarrow0}\dfrac{f_2(x+h)-f_2(x)}{h}=\lambda\lim_{h\rightarrow0}\dfrac{f_1\left(\frac{x+h}{\lambda}\right)-f_1\left(\frac{x}{\lambda}\right)}{h}.\: $Since$ f_1 $is continuously differentiable we see $ f_2 $ is differentiable and its derivative is continuous and hence $f_k$. $\Bigg)$.  Further since $f_1(0)=0$. So I assumed $f_1(x)=x\cdot g(x)$, where $g(x)$ is continuously differentiable and $g(0)\neq0$. Assume $x\neq0,\:f_2(x)=\lambda\cdot\dfrac{x}{\lambda}g\left(\dfrac{x}{\lambda}\right)=x\cdot g\left(\dfrac{x}{\lambda}\right)\implies f_3(x)=x\cdot g\left(\dfrac{x}{\lambda^2}\right)$ $\implies f_k(x)=x\cdot g\left(\dfrac{x}{\lambda^{k-1}}\right)$.   So taking limit as $k\rightarrow \infty$, we get $\displaystyle \lim_{k\rightarrow\infty}f_k(x)=x\cdot g(0)$. Now I am not sure how to proceed or whether there is something wrong here. Please provide hints or suggestions.  $\rule{17cm}{1pt}$  To sum up (using the arguments made by $\textbf{zhw}$ and $\textbf{Paul Frost}$)  When $\displaystyle x\neq0, \lim_{n\rightarrow\infty}f_{n+1}(x) = \lim_{n\rightarrow\infty}x\,\frac{f_1(x/\lambda^n)}{x/\lambda^n} = \lim_{n\rightarrow\infty}x\,\frac{f_1(x/\lambda^n)-f_1(0)}{x/\lambda^n}=x\:f'(0).$ Thus  $$f_n\rightarrow\begin{cases}0&amp;x=0\\x\:f'_1(0) &amp;x\neq0\end{cases}=f $$ Did something change in the behavior of blocking users' votes if they don't have sufficient rep to vote? I've just noticed that if you try to vote without having a privilege to vote, you're greeted with such message:        So, that's interesting. Does it mean that if I gain 125 (or 15, in case I'm upvoting not having 15 reps) rep on that site, my "suggested" vote goes public and until then it would be recorded as it's done with anon feedback?  And until this change, the votes weren't recorded as anonymous, or has the new message saying that it acts as feedback been added just to not upset users?