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    (Original post by Occams Chainsaw)
    I've got a good log question that my maths lectures couldn't/wouldn't answer for me so I'm wondering if you can/will please!

    Why do negative base values sometimes work but sometimes not? I presume it's got something to do with complex numbers but I can't make the connection tbh.
    erm, let me go and think on that one

    Sooooo many things annoy me about A Level maths like this! Like why the CoM of a semi-circle is 4(r)/3(pi)
    another erm ... though I do know how to derive CoMs somewhere at the back of my mind

    and why exactly the way we integrating (namely dividing my the new power) actually works.
    Do you mean as an anti-derivative or why it gives area through that

    Or how sine is actually derived (not that circle on an axis thing, that's cheating!).
    The trig functions occur naturally like pi ... they are just ratios

    Mechanical use of the principles is stupid.. We need to know why, I think! Maybe I should be able to work it out for myself but I am far too lazy and probably not clever enough to do it!
    stop asking hard questions in the holidays
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    (Original post by Indeterminate)
    Well, to answer your main Q

    \log_{-3} 9 = 2

    works but it's not of any use tbh
    \log_{-2} 8?
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    (Original post by TenOfThem)
    Do you mean as an anti-derivative or why it gives area through that
    I just mean that I don't really understand how you actually derive that using the steps of integrating will work. It makes sense to reduce the power but dividing by the new power seems anti-intuitive to me. It obviously works but I don't know how! Maybe I need to study real analysis? I asked my lecturer who just said "it's the opposite of differentiation". Although that is true, it didn't really answer my question!
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    (Original post by Occams Chainsaw)
    \log_{-2} 8?
    hmmm

    \log_{-2}(-8)
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    (Original post by Occams Chainsaw)
    \log_{-2} 8?
    It's all well and good saying something works by the "face value" definition of it, but we also have to consider everything that goes with it.

    For example

    \log_{-3} 9 = \dfrac{\ln 9}{\ln -3}

    So it's pretty pointless to think of negative bases.
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    (Original post by Eilidh.DearX)
    Okay, so here is my working:

    2log3 y - log3(y+4) = 2
    log3 y^2 - log3(y+4) = 2
    log3( y^2 / y+4 ) = 2
    Take 3 to the power of both sides to give
    3^ (log3 y^2 / y+4 ) = 3^2
    y^2 / y+4 = 9
    multiply through by y + 4
    y^2 = 9 (y+4)
    y^2 = 9y = 36
    y^2 - 9y -36 = 0
    (y - 12) (y + 3) = 0
    so y = 12, -3
    can't do a negative number with logs, so y = 12 is the answer

    Hoped this helped
    you can't post full solutions! instead you have to guide the OP to the answer; read this: http://www.thestudentroom.co.uk/wiki...elp_Guidelines
    btw, it wasn't me who negged you.
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    (Original post by TenOfThem)
    hmmm

    \log_{-2}(-8)
    I actually just tried plotting the graph of my question and realised you don't get a line. Just a series of points. Perhaps it's something to do with that? I don't know!
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    (Original post by Indeterminate)
    It's all well and good saying something works by the "face value" definition of it, but we also have to consider everything that goes with it.

    For example

    \log_{-3} 9 = \dfrac{\ln 9}{\ln -3}

    So it's pretty pointless to think of negative bases.
    I was just interested! Anomalies bug me!
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    (Original post by Occams Chainsaw)
    I was just interested! Anomalies bug me!
    Reminds me of myself a bit :lol: Eventually, I sort of settled down.

    By the way,

    \log_{-3} 9 = \dfrac{\ln 9}{\ln - 3} = \dfrac{\ln 9}{\ln 3 + i\pi}

    using the definition that

    \ln(-1) = i\pi

    which is, in fact, a consequence of Euler's identity.
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    (Original post by nm786)
    you can't post full solutions! instead you have to guide the OP to the answer; read this: http://www.thestudentroom.co.uk/wiki...elp_Guidelines
    btw, it wasn't me who negged you.

    Oh, oops! I didn't know you weren't supposed to. But how can you help people if you have to 'dance round answers as it were'? :confused:
    Don't worry I wasn't going to assume you did, I haven't even looked at my post, must have loads of negs lol xD Am I going to get into trouble?
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    (Original post by Occams Chainsaw)
    Why do negative base values sometimes work but sometimes not? I presume it's got something to do with complex numbers but I can't make the connection tbh.

    Sooooo many things annoy me about A Level maths like this! Like why the CoM of a semi-circle is 4(r)/3(pi) and why exactly the way we integrating (namely dividing my the new power) actually works. Or how sine is actually derived (not that circle on an axis thing, that's cheating!).
    These are all really good questions, and certainly not things you should be able to work out by yourself (unless you are descended from Newton!)

    For any base, we usually define
    a^x \equiv e^{xlog a} where I'm using log to denote the natural (base e) logarithm. So think about how this will work when a is negative! (and yes, it does involve complex numbers)

    From a practical point of view, think about how you graph a^x for a real number a. If a is positive then you get a nice smooth curve, you can work out the answer for integer x directly, and interpolate the points in between (for example if x is rational, you know how to work out xth roots etc). However, how would you work out (-3)^{2.635}?
    By the laws of indices (-3)^{2.635} = (-3)^2 \cdot (-3)^{0.635}, but how are you going to evaluate that 2nd factor? Again, this leads you into the realms of complex numbers.

    For the centre of mass, I'm not quite sure what you're asking. The CoM has to be something - if it were 7pi/43 you'd be asking why it was that value! Or are you asking how to get the answer for the CoM? This is done by integration - typically explained in M3 (at least in the old Edexcel book I have).

    Integration and powers? Simple answer: if you define indefinite integration as the reverse process to differentiation, and you know that differentiation brings down a power and reduces the power by 1, then you can see why integration must involve division by (power+1). Deeper answer: if you didn't know anything about differentiation but defined (definite) integration as the area under a curve, then it's still possible to get an answer for the area under x^n but it takes more work and a knowledge of infinite sums and limits.

    Definition of sine? Well an analyst defines it by an infinite power series and then proceeds to show that it satisfies all the properties you'd expect of it!
    The school method starts with the definition as a ratio of sides in a right-angled triangle, but clearly this only makes sense for acute angles. So then you have to extend the definition to be the y-coordinate of a point moving round the unit circle - this gives you your periodicity etc. There's a geometric argument you can use to show that sinx/x -> 1 as x->0 and this gives you your derivative etc.
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    (Original post by Eilidh.DearX)
    Oh, oops! I didn't know you weren't supposed to. But how can you help people if you have to 'dance round answers as it were'? :confused:
    Don't worry I wasn't going to assume you did, I haven't even looked at my post, must have loads of negs lol xD Am I going to get into trouble?
    You won't "get into trouble" - unless you do it repeatedly

    Most people who post in the Maths forum are either (a) lazy and want the answers given to them, but they go away quickly when they realize that isn't going to happen; or (b) stuck at some point and in need of a hint or a confidence boost. If they're given a bit of a nudge in the right direction they can usually go on by themselves and solve the problem!!
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    (Original post by davros)
    You won't "get into trouble" - unless you do it repeatedly

    Most people who post in the Maths forum are either (a) lazy and want the answers given to them, but they go away quickly when they realize that isn't going to happen; or (b) stuck at some point and in need of a hint or a confidence boost. If they're given a bit of a nudge in the right direction they can usually go on by themselves and solve the problem!!
    Aw right okay, thanks!
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    (Original post by davros)
    These are all really good questions, and certainly not things you should be able to work out by yourself (unless you are descended from Newton!)

    For any base, we usually define
    a^x \equiv e^{xlog a} where I'm using log to denote the natural (base e) logarithm. So think about how this will work when a is negative! (and yes, it does involve complex numbers)

    That makes perfect sense now. I hadn't thought about that definition of a^x before. It's actually pretty interesting!
    From a practical point of view, think about how you graph a^x for a real number a. If a is positive then you get a nice smooth curve, you can work out the answer for integer x directly, and interpolate the points in between (for example if x is rational, you know how to work out xth roots etc). However, how would you work out (-3)^{2.635}?
    By the laws of indices (-3)^{2.635} = (-3)^2 \cdot (-3)^{0.635}, but how are you going to evaluate that 2nd factor? Again, this leads you into the realms of complex numbers.

    I did notice the graph didn't form a line, but rather a scattering of points.

    For the centre of mass, I'm not quite sure what you're asking. The CoM has to be something - if it were 7pi/43 you'd be asking why it was that value! Or are you asking how to get the answer for the CoM? This is done by integration - typically explained in M3 (at least in the old Edexcel book I have).
    I wondered why it was 3r/4pi. I know it had to be something but it just didn't look right when I first saw it. That may sound a little arrogant but usually I can happily see that something is right/wrong intuitively -- in A Level maths, anyway. That probably isn't something I should rely on but it does help! Integration? I tried doing it in ratios but it got very complicated very fast!

    Integration and powers? Simple answer: if you define indefinite integration as the reverse process to differentiation, and you know that differentiation brings down a power and reduces the power by 1, then you can see why integration must involve division by (power+1). Deeper answer: if you didn't know anything about differentiation but defined (definite) integration as the area under a curve, then it's still possible to get an answer for the area under x^n but it takes more work and a knowledge of infinite sums and limits.
    What I am interested in is why when you intergrate, say x dx you then divide by the new power, in this case, 2. I know it's the opposite of differentiation but then I don't know why you multiply in that either, so it doesn't really help!
    Definition of sine? Well an analyst defines it by an infinite power series and then proceeds to show that it satisfies all the properties you'd expect of it!
    The school method starts with the definition as a ratio of sides in a right-angled triangle, but clearly this only makes sense for acute angles. So then you have to extend the definition to be the y-coordinate of a point moving round the unit circle - this gives you your periodicity etc. There's a geometric argument you can use to show that sinx/x -> 1 as x->0 and this gives you your derivative etc. The point about the acute angles is why I got confused. It didn't make sense! In honesty, I'm going to have to read over that definition again and play around before I understand it but I appreciate the answer!
    Thank you so much! If only my lecturers at college could give such good answers. I suppose they have to worry about the rest of the class being able to understand what's strictly on the syllabus before satisfying my curiosity!
    I hope you don't think it's rude to comment within your quote, sort of like how a teacher would mark an essay, because I think it makes a response so much more coherent!

    Again, I really appreciate the help!
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    (Original post by Occams Chainsaw)

    Thank you so much! If only my lecturers at college could give such good answers. I suppose they have to worry about the rest of the class being able to understand what's strictly on the syllabus before satisfying my curiosity!
    I hope you don't think it's rude to comment within your quote, sort of like how a teacher would mark an essay, because I think it makes a response so much more coherent!

    Again, I really appreciate the help!
    No problem - it's good that you're asking questions; more people should do this.

    The problem with the modern modular exam system is that you're permanently in a cycle of preparing for exams, taking exams, reviewing exams etc, so to be fair to teachers most of what they do now is highly constrained by the need to hit targets and teach to a test. I was fortunate enough to do fewer exams and read more outside the syllabus when I was at school.

    If you've looked at differentiation from first principles, that should answer your question about multiplying by powers when you differentiate a power of x.

    If I think of any better answers to your question I'll come back
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    (Original post by Occams Chainsaw)
    I've got a good log question that my maths lectures couldn't/wouldn't answer for me so I'm wondering if you can/will please!

    Why do negative base values sometimes work but sometimes not? I presume it's got something to do with complex numbers but I can't make the connection tbh.
    If x=log_(-b) (a), and b>0, then (-b)^x=a. Thus, x is only defined when a is an even power of b or the negative of an odd power of b?

    EDIT: ninja'd
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    (Original post by davros)
    These are all really good questions, and certainly not things you should be able to work out by yourself (unless you are descended from Newton!)
    the apple does not fall far from the tree...
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    (Original post by davros)
    No problem - it's good that you're asking questions; more people should do this.

    The problem with the modern modular exam system is that you're permanently in a cycle of preparing for exams, taking exams, reviewing exams etc, so to be fair to teachers most of what they do now is highly constrained by the need to hit targets and teach to a test. I was fortunate enough to do fewer exams and read more outside the syllabus when I was at school.

    If you've looked at differentiation from first principles, that should answer your question about multiplying by powers when you differentiate a power of x.

    If I think of any better answers to your question I'll come back
    If everyone asked these sorts of questions then teachers would probably have a very hard time!

    As a maths undergraduate, did you have any opportunities to look at things like Game Theory when you were completing your degree? Right now I am looking at physics (because it's cool in general) and economics (because I love game theory) but figured that maths might be a good way to combine the two in their most theoretical forms! STEP is going to be a pain though! :/
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    (Original post by Occams Chainsaw)
    If everyone asked these sorts of questions then teachers would probably have a very hard time!

    As a maths undergraduate, did you have any opportunities to look at things like Game Theory when you were completing your degree? Right now I am looking at physics (because it's cool in general) and economics (because I love game theory) but figured that maths might be a good way to combine the two in their most theoretical forms! STEP is going to be a pain though! :/
    I must admit that games theory, optimization and the suchlike never really appealed to me. After my 1st year I mainly focused on theoretical physics courses, although I did attend number theory lectures as that always appealed to me. Ironically, I've now forgotten almost all the applied stuff I learnt owing to lack of practice, and I've started rereading stuff like analysis which I learnt at a hectic pace and never really appreciated!
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    (Original post by davros)
    I must admit that games theory, optimization and the suchlike never really appealed to me. After my 1st year I mainly focused on theoretical physics courses, although I did attend number theory lectures as that always appealed to me. Ironically, I've now forgotten almost all the applied stuff I learnt owing to lack of practice, and I've started rereading stuff like analysis which I learnt at a hectic pace and never really appreciated!
    Did you go to Cambridge, by any chance? It just seems that a lot of the maths course there is geared for theoretical physics!
    Well, if I could have a split between physics and game theory-type stuff in the second year then that would be cool. I could then probably specialise in the third!
    I've found the same as you with simple statistics etc that I had learnt at GCSE. I realised I've forgotten everything I learnt about probability which probably isn't very good!

    Analysis and topology look pretty difficult! Because I don't see any real application it doesn't appeal to me as much as physics where I can learn something and say "oh, so that's how that works". I think that is more satisfying for me.
 
 
 
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