I am having some trouble solving this equation for $x$: $$a^x + b^x = c.$$ I thought I should use logs but am unable to.
Well the question is to solve for $x$ in $$a^x+b^x=c \tag{a,b,c are constants}$$   Well as of me, I tried to put $\ln{}$ on both sides which does not seem to help. Apart from this I don't seem to have any other way to solve it.   I turned to WolframAlpha and the solution which was given was : $$x=\dfrac{-i(2\pi n +\pi)}{\ln{a}-\ln{b}} , {n \in \mathbb{Z}}$$  And if $n=0$ $$x=\dfrac{-i\pi}{\ln{a}-\ln{b}}$$  The only thing I am able to recognize is $e^{i\pi}=1$ ... i guess is used.  Link :   So I see that $x$ is independent of the constant $c$.  So how to solve it? Please help...Thanks!    EDIT - It seems that WolframAlpha is taking $c=0$ as suggested by Donkey_2009. And when i write "Solve for x in a^x + b^x = k" it is not returning anything. So are you guys aware of a method to solve this? Also are you guys aware of any other website/software like WA?    P.S. - Also I see that IF we know the values of $a,b,c$ resulting in something like $2^x+3^x=100$ , then Newton's method is the only method I am aware of that can solve it...  Any other suggestions?
The mystery of the tagged answer - and more It is (currently)   to tag answers. Yet, if I query , then  does have a tag, namely javascript. I'm at a loss: how did that happen?&dagger;  SELECT Id, Tag FROM Posts, PostsTags  WHERE Posts.PostTypeId = 2 AND Posts.Id = PostsTags.PostId   Edit: The are also five answers (, , ,  and ) with a view count of 1 instead of 0:  SELECT Id, ViewCount FROM Posts WHERE PostTypeId = 2 AND ViewCount &lt;&gt; 0   These results also hold through at least January 10.    &nbsp; &dagger;: PostTypeId = 2 means that the post is an answer, not a question (PostTypeId = 1). SPWeb.getFile(Url) exception : value does not fall within the expected range I received $1000 and was asked to send it back. How was this scam meant to work? Possible Australianism - netbank = a website + app by which an Aussie can do their banking, including transferring money to other people.  A while ago, I ended up with $1000 transferred into my account, which in the description looked like a netbank transfer, not like a cheque, cash deposit etc, and which looked like it had already cleared and landed in my account. Usually when something is still working through the bank's system it displays as 'pending', but did not in this case.  I received a phone call from someone, saying that they had accidentally typed in the wrong account number, and wanting me to send them the same amount of money back via netbank.   Surely bank account numbers have a checksum, which make it relatively difficult for a typo to result in a payment going to the wrong person? What are likely sources of them being able to find my phone number to call me? Obviously, the bank either being hacked, or giving it out in response to a query from this person (but surely the bank is unlikely to tell me my own mobile number if I claim to have forgotten it?). Presumably either my phone, or the phone of someone I have sent money to/received money from could be compromised. Has anyone heard of any instances of this happening?  I then went to my bank, explained the situation and said that I did want to refund the money through the officially sanctioned methodology. After the laughter died down, the bank told me to ask the person to fill out a form, then they would get me to fill out a form that would authorize undoing that exact payment.  I rang up the person to tell them this, and over the course of about a week received an increasingly elaborate story about how that wasn't working, they needed the money immediately, there was a fee to fill out that form and they couldn't afford to pay it etc.  After that, I never heard back from them again. I check my netbank for several weeks after, expecting the money to magically disappear, but it did not. This was probably a year ago, and I'm pretty vague with money so may not have noticed if it disappeared more recently. How were they expecting to benefit?  Were they hoping to get the transfer from me, and undo the transfer to me, to leave them $1k up? If so, how can an internet banking transfer be 'undone' once it has already shown as 'received'. Or was this to do with money laundering? The account they requested me to send money to was the same as the one that the money appeared to come from. This was a scam, right? Recreate plot when data file changes Is it possible to use the TikZ/external library and have pgfplots recreate a figure when a data file that is plotted changes? Say I have something like the following:  file.tex:  \documentclass{article}  \usepackage{tikz,pgfplots} \usetikzlibrary{external} \pgfplotsset{compat=newest} \tikzexternalize \newcommand{\figFilename}{}  \begin{document}   \begin{figure}[t]     \centering     \renewcommand{\figFilename}{thePlot}     \tikzpicturedependsonfile{\figFilename.tikz}     \tikzsetnextfilename{\figFilename}     \input{\figFilename.tikz}     \caption{The caption.}   \end{figure} \end{document}   thePlot.tikz:  \begin{tikzpicture}   \begin{axis}[     xlabel={$x$},     ylabel={$y$},]     \addplot+[only marks,mark=x]       table[x=x,y=y] {data.dat};   \end{axis} \end{tikzpicture}%   data.dat:  x y 0 20.44 1 20.06 2 19.76 3 19.54 4 19.40 5 19.34   If I change thePlot.tikz, the external library will detect that the figure is no longer up-to-date and recreate it. Is it possible to have the same effect when any of the plotted data files changes as well? I tried something along the lines of \tikzpicturedependsonfile{data.dat}, and it doesn't work. That wouldn't be an ideal solution anyway, as most times I use many data files in each plot and listing all dependent files manually would be more pain than gain.  So is there an easy, automated way for the external library to check all data files included in a tikzpicture and recreate the plot if at least one of them has changed? Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$ I'm undergraduate student of mathematics.  I need to prove: $$\sum_{k=0}^{n} \binom{2n}{2k}= 2^{2n-1}$$ Can you please help me iOS 10 remember to "not trust this computer" Fluid filled harmonic oscillator A vessel (preferably circular) filled with water is accelerating unidirectionally such that the level of water is higher on one end than the other. What I want to know is that if the vessel is immediately stopped, the water level will force itself back to the equilibrium point, but in doing so push the other end up. If water is an ideal fluid, what type of oscillation will occur in this situation.   I can tell that maybe the mathematical expression for this oscillation will be complex, but please do let me know if I am thinking this correctly.   I am considering, finding the oscillation equation for infinitesimally small u-tubes(considering a 2-d plane) and integrating it to get the final expression.  Thanks in advance How to add a space between a string for every 5th letter? Ideal contained in a finite union of prime ideals Let $I \subset R$ be an ideal and $P_i$  $(i=\{1,...,n\})$ prime ideals with $I\subseteq\bigcup_{i=1}^nP_i$. Prove that then $I$ is contained in one $P_i$.   I don't know how to show this because I don't have any approach.  So I am looking for something to start with or something like a sequence of tips (Since I know this is a large proof)  Thanks in advance! Movie with killer androids (glowing eyes, screaming) Allow us to vote to delete from the Android app Currently, the only way to vote to delete (and undelete, since you can't see deleted posts) in the Android app is to open the post in a browser, and then vote to delete from there.  Can we get delete vote functionality within the app, as well as the ability to delete our own posts?  Worth to mention: users , so it's possible in the API.  Related: What are the guidelines for reviewing? About reviewing The review queues contain posts that possibly need further action from the community, like improvement, closure or deletion. By presenting those posts for peer-review, the system aims for higher quality content. Some user actions, such as suggesting an edit, casting a flag, close vote or reopen vote, or posting for the first time, can trigger the inclusion of a post on a review queue. There are also posts included on the queues algorithmically, such as the contents of Late Answers and Low Quality Posts. General guidelines  Always read the full post you are reviewing. Don't rush. Take the necessary time to read the post carefully. If you don't care about a post, just click Skip or Not sure. When you need more context, open the post link to see the question and all answers. Most posts coming up for review can be improved. Use the Edit option, and edit thoroughly. If you're unsure how to review a post (perhaps it's outside your areas of expertise), skip it. Someone who understands it better will review it later.  Guidelines for specific review queues         (Stack Overflow only) I have a large number of zip files that were compressed using the zip command. I would like to recompress them with the -9 flag to improve the compression ratio. Does anyone know if that can be done without manually decompressing and then compressing.  PS. I need to keep them as zip files since they are served to windows users( and as such have white spaces in their names) No help from ?-command Unparsed HTML shows on search result page How do I use sudo to redirect output to a location I don't have permission to write to? I've been given sudo access on one of our development RedHat linux boxes, and I seem to find myself quite often needing to redirect output to a location I don't normally have write access to.  The trouble is, this contrived example doesn't work:  sudo ls -hal /root/ &gt; /root/test.out   I just receive the response:  -bash: /root/test.out: Permission denied   How can I get this to work? Use of very as an adjective is (in my experience) most frequently attested in phrases like     ...the very person I was looking for.   To use adjective very with the indefinite article sounds quite bizarre in this case:     ?...a very person I was looking for.   But consider this sentence, where indefinite a is used:     Visualizing a very transformation of the master-slave relation, Thomas presses further: the slave is brother to the master, even doubly a brother.   OED treats the two uses as separate senses of the word, with the following definitions:           (first use):&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;With limitation (usually expressed by the or a possessive) to particular instances      (second use): Really or truly entitled to the name or designation   Here are two respective examples quoted in OED:     What would you say to me now, and I were your verie, verie Rosalind? (1616)   Thence we went into Queen Mary's room, and saw that beautiful portrait—that very queen and very woman. (1857)   Clearly, the difference in meaning between the two senses is quite light, and it seems to me like a kind of difference which is recurrent enough to have its own term. What is the proper term? Are there other English words having similarly related pairs of senses? And lastly, is OED jumping the gun in treating these as two senses? Could the difference in meaning instead be due to the choice of determiner? Why an abstract class implementing an interface can miss the declaration/implementation of one of the interface's methods? A curious thing happens in Java when you use an abstract class to implement an interface: some of the interface's methods can be completely missing (i.e. neither an abstract declaration or an actual implementation is present), but the compiler does not complain.  For example, given the interface:  public interface IAnything {   void m1();   void m2();   void m3(); }   the following abstract class gets merrily compiled without a warning or an error:  public abstract class AbstractThing implements IAnything {   public void m1() {}   public void m3() {} }   Can you explain why? If I wanted to say something occurred occurred between 1627 and 1629, I would use &quot;the late 1620s&quot;. However, if I wanted to say something occurred between 1607 and 1609, I couldn't use &quot;the late 1600s&quot; because that would imply it occurred between 1670 and 1699. Attacks of the MAC construction $\mathcal{H}(m\mathbin\|k)$ for common hashes $\mathcal{H}$? Consider a common practically-collision-resistant hash function $\mathcal{H}$ (e.g. , , , ), perhaps based on the  as are the first three. We define a  $\mathcal{C}$ $$(k,m) \mapsto \mathcal{C}(k,m)=\mathcal{H}(m\mathbin\|k)$$ where $\mathbin\|$ denotes concatenation, $k$ is a secret key (constant, or at least of fixed size), and $m$ is a message (possibly of variable length). Assume that an adversary can (iteratively) submit queries with $m_j$ and obtain $C(k,m_j)$, and wants to obtain $k$ or otherwise compute $C(k,m)$ for some $m\ne m_j$.  That MAC $\mathcal{C}$ is not trivially bad. In particular, if $\mathcal{H}$ was indistinguishable from a random function in the , $\mathcal{C}$ would be secure. And even though $\mathcal{H}$ may have the , it does not turn into a devastating attack on $\mathcal{C}$.  The less impractical generic attack that I see is that if a collision was known for $\mathcal{H}$ with the colliding messages of moderate identical length, it could be deduced countless collisions for $\mathcal{C}$. Hence security is demonstrably not better than collision-resistance of $\mathcal{H}$ (for identical-length messages). We could assume that $k$ is half the size of the result of $\mathcal{H}$, and hope that the security is about  or  or , 280, 2128, 2256 hash rounds for SHA-1, RIPEMD-160, SHA-256, SHA-512.   What are the known attacks against $\mathcal{C}$ (better than the above), and their cost, for each of these common hashes?  Is there hope for an argument that an attack against $\mathcal{C}$ would turn into an attack of similar cost against $\mathcal{H}$, or hint of the contrary?  Update: this  is of interest, but I fail to find that it really answers the present question.  Update 2: I am aware that the construction considered is weaker than , and in particular is vulnerable to collision on $\mathcal H$; I stated that, and that it is thus pointless to have the key wider hopeless to target security against some attacks better than half the hash's size. I'm asking exactly what cryptanalytic attack better than finding a collision on $\mathcal H$ there are. There is room for such an attack only by exploiting a weakness in the structure or/and the round function of a concrete $\mathcal H$.