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Thread: Not so smart question about exponential stuff

  1. #1 Not so smart question about exponential stuff 
    Member pienapple27's Avatar
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    I decided to make a new thread just because the other one said "linear stuff" and this no longer fits in that.
    I was just looking at y=2^x (I have no idea how to superscript x, help?) and then I started to draw a graph in my head and I noticed that when x is negative y is almost but not quite zero, and it gets closer to zero as I go further in the -x direction, but does it ever reach zero? Is this what you call a limit? As in: as x tends to -infinite, y tends to zero or something?

    Also, how can I solve for x, I mean, turn it from y=a^x to x=something?
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  2. #2  
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    Quote Originally Posted by pienapple27 View Post
    I decided to make a new thread just because the other one said "linear stuff" and this no longer fits in that.
    I was just looking at y=2^x (I have no idea how to superscript x, help?)
    In forum BB code, you can use the [sup]...[/sup] tags, as in y = 2[sup]x[/sup].

    Quote Originally Posted by pienapple27 View Post
    and then I started to draw a graph in my head and I noticed that when x is negative y is almost but not quite zero, and it gets closer to zero as I go further in the -x direction, but does it ever reach zero? Is this what you call a limit? As in: as x tends to -infinite, y tends to zero or something?
    That's the right idea. You can get as close to zero as you want by going far enough in the negative-x direction, but you can never actually reach zero (every time you reduce x by 1, the value of 2x is just divided by two, so it could never end up at zero unless it was zero to begin with, which it wasn't). In mathematical jargon, "the limit of 2x as x tends to negative infinity is zero", written like so:


    Sometimes people write


    (read "2x tends to zero as x tends to negative infinity"), which means exactly the same thing.

    Quote Originally Posted by pienapple27 View Post
    Also, how can I solve for x, I mean, turn it from y=a^x to x=something?
    You want a logarithm for that job. If you know that , then you can write (read "x equals log to the base a of y"). Scientific calculators usually have functions for logarithms to base 10 and base e = 2.718... (the so-called natural or Napierian logarithm, often denoted by rather than ); from those you can work out logs to any other base using a formula such as this:


    or this, which produces exactly the same result:

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    Well, I'd like to, first of all, thank you, btr for having the patience to explain and answer all my questions
    I think I just learned a new thing! log of a number is the exponent of the base to make that number --> log2(8)=3 because 23=8 yay!
    Also, I've been reading tons of wikipedia pages but they are kinda hard to understand. What is "e"? What's so special about it? Why is it the base of the natural logarithm? Why is it called natural logarithm? Maybe I'm repeating myself, but what is so special about loge that it has to be distinguished from other bases?
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    When you do integration it becomes important as the integral of 1/x = ln x where x= e ^ ln x

    It's also very special because e^ i pi = -1

    If you stick with maths all this should become clear when you hit 18.


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    inte-gra-ti-on? hm maybe you think I'm crazy asking this but could you be so kind as to explain calculus to me? xD
    I want to learn, it's just wikipedia only has advanced definitions of stuff and equations I don't have the knowledge to comprehend and my maths teacher doesn't help :c
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    I did manage to explain it to my brother when he was 12, but I knew exactly what level he was at. There is lots of stuff that must be covered first otherwise it just won't make sense. Learning maths is like building a house, you have to get the foundations in first and give them enough time to dry. You could try googling easy calculus and see how far you can get. If you then have a specific question please post it.
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    Okee dokee!
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    btr
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    Another way of seeing how e arises is this.

    If you plot the graph y = ax for some fixed constant a > 1, you get something like this:


    You can see the curve crosses the y-axis at (x, y) = (0, 1). If we zoom right in on that point the curve looks like this:


    It looks almost like a straight line, right? And if you zoom in more and more, it looks even more and more like a straight line. You can imagine, perhaps, that we zoom in infinitely far and get something which is exactly a straight line.

    The question which then arises is what is the gradient of that line? Well, the answer depends on the value of a. If a = 2 the gradient is approximately 0.6931. If a = 3 the gradient is approximately 1.0986. Larger values of a give larger gradients.

    Hmm, that's interesting. It looks like we could find a value of a somewhere between 2 and 3 for which the gradient is exactly 1. And we can - the value turns out to be e!

    So, that is one way in which the number e is so special, and probably one of the first ways you will learn about when you come to study calculus.
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    The gradient is the value you get through differentiation right, aka slope?
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  10. #10  
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    That's exactly it.
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  11. #11  
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    so e is the magic a that gives is a unity slope
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  12. #12  
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    Precisely.

    You can generalise that slightly, it turns out: the curve y = ex is the unique curve that simultaneously meets the following criteria:

    1. y = 1 when x = 0.

    2. The gradient of the curve at the point (x, y) is equal to y.
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