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Math Help - Exponential Functions and Logarithms?

  1. #1
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    Exponential Functions and Logarithms?

    Where's the best site to understand this topic? I need help and can only understand some parts of it.

    For example for my homework given by my teacher I must simplify giving the answer with positive index:

    a^2 b^3 / ab X a^2 b^5 /a^2 b^2

    and another question:
    a^1/4 X a^2/5 X a^-1/10

    and questions like:
    write 2^n X 8^n / 2^2n X 16 in the form 2^an+b

    There's lots more like these then to the sketch graphs which want us to give equations of asymptopes and the axes intercepts etc. Transformations..

    Any recommended website that teaches you this in detail?

    Thanks.
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  2. #2
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    For all the problems you have to use

    a^ma^n = a^{(m+n)} and \frac{a^m}{a^n} = a^{(m-n)}
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  3. #3
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    How do we use it though?
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  4. #4
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    I'll show you the first one:

    \frac{a^2 b^3}{ab} \times \frac{a^2 b^5 }{a^2 b^2}

    First, cross out those which cancel out.

    Rearrange this way to enable you to see the powers:

    \frac{a^2 b^3}{a^1b^1} \times \frac{a^2 b^5 }{a^2 b^2}

    Apply what sarigama told you:

    a^{(2-1)} b^{(3-1)} \times a^{(2-2)}b^{(5-2)}

    a^1 b^2 \times a^0b^3

    Reapply, but now with multiplication instead of division:

    a^{(1+0)} b^{(2+3)}

    a^1 b^5
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  5. #5
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    Another way to see this is to write \frac{a^2b^3}{ab} as \frac{aabbbb}{ab}. Now you can cancel the "a" and "b" in the denominator with one "a" and "b" in the numerator leaving \frac{abb}{1}= ab^2. That's exactly where the \frac{a^2b^3}{a^1b^1}= a^{2-1}b^{3-1}= a^1b^2= ab^2 comes from.

    Similarly, the other fraction, \frac{a^2b^5}{a^2b^2} can be written as \frac{aabbbbb}{aabb} and now we can cancel the "aa" and "bb" in the denominator with two of the "a"s and "b"s in the numerator leaving \frac{bbb}{1}= b^3

    Putting those together, \frac{a^2b^3}{ab}\frac{a^2b^5}{a^2b^2}= (ab^2)(b^3)= a(bb)(bbb) so there are 1 "a" and 5 "b"s: ab^5
    Of course, it is easier to think of subtracting and adding the exponents than it is to write all of the "a"s and "b" separately and count them but that is what is really happening.
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  6. #6
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    Quote Originally Posted by HallsofIvy View Post
    Of course, it is easier to think of subtracting and adding the exponents than it is to write all of the "a"s and "b" separately and count them but that is what is really happening.
    That is what I really needed to understand. What the process was. Thank You!

    Could you please explain the same way for this one:

    (2a)^2x8b^3
    -------------
    16a^2b^2

    And also how will we expand 16(ab)^2 ? If it was in the equation...
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  7. #7
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    Quote Originally Posted by Alive View Post
    That is what I really needed to understand. What the process was. Thank You!

    Could you please explain the same way for this one:

    (2a)^2x8b^3
    -------------
    16a^2b^2
    Since both 2 and a are in the parentheses in (2a)^2 both are squared: (2a)^2= 4a^2
    \frac{(2a)^2(8b^3}{16a^2b^2}= \frac{4a^2(8b^3)}{16a^2b^2}. (I am assuming that "x" in the numerator is a multiplication. It's not a good idea to use "x" to mean multiplication in algebra problems like this. Use parentheses where necessary.)

    Putting together same "bases", we have \frac{4(8)}{16}= \frac{32}{16}= 2, \frac{a^2}{a^2}= a^{2- 2}= a^0= 1, and \frac{b^3}{b^2}= b^{3-1}= b^1= b
    \frac{(2a)^2(8b^3)}{16a^2b^2}= 2b

    And also how will we expand 16(ab)^2 ? If it was in the equation...
    Again the a and b are inside parentheses so they are both squared: 16(ab)^2= 16a^2b^2

    By the way, although these problems involve exponents, they have nothing to do with "exponential functions" or "logarithms" though perhaps that is next in your course. Something like 2^a is an "exponential function". a^2 is not.
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