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Math Help - Proof of Mercatorís deduction of logarithms from a hyperbola functions

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    Proof of Mercatorís deduction of logarithms from a hyperbola functions

    For my History of Maths class I received the following challenge:

    Prove geometrically that the area under the hyperbola (y=1/x) from 1 to p, is equal to the area from q to pq. Deduce that the area (as function) has the addition property of logarithms, log(ab)=log(a)+log(b).
    The problem (to me!) that it needís to be done geometrically; something to do with ratioís.

    Of course the area of a rectangle formed by op*ppí (let pí be 1/p) equals oq*qqí and opq*pq(pq)í
    But Iím not getting any further staring at my hyperbola...
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    Super Member ILikeSerena's Avatar
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    Re: Proof of Mercatorís deduction of logarithms from a hyperbola functions

    Hey TDA120!

    If you have a rectangle with sides w and h, then you can stretch that rectangle and still have the same area.
    The stretched rectangle would for instance have sides qw and h/q.

    The definition of a (Riemann) integral is the summed surface area of infinitely many rectangles with height f(x) and width dx...
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    Re: Proof of Mercatorís deduction of logarithms from a hyperbola functions

    But wonít I be using integrals already then by using the log rules;
    lnpq -lnp=lnp-ln1
    lnp+lnq-lnp=lnp-0
    lnq=lnp

    Because this is exactly what I shouldnít be doing...
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    Super Member ILikeSerena's Avatar
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    Re: Proof of Mercatorís deduction of logarithms from a hyperbola functions

    The difference would be that you do not assume that ln(x) is the area under the graph with all the properties it has.

    You would assume there is some unknown function a(x) that describes the area under the graph.
    Geometrically you deduce some of its properties, typically with inscribed rectangles.
    You can stretch those rectangles to another part of the graph, deducing that the area must be the same.

    And then you can conclude that this unknown function a(x) has certain properties related to scaling geometric figures.
    And hey, these properties will turn out to be exactly the properties from the ln(x) function.
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    Re: Proof of Mercatorís deduction of logarithms from a hyperbola functions

    If you approximate the first area by a suitable summation of n rectangles, you can easily prove that the corresponding summation for the second area gives the same result. The rectangles' heights are multiplied by 1/q, and their widths are multiplied by q. Let n go up, and bingo. Interesting question!
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    Re: Proof of Mercatorís deduction of logarithms from a hyperbola functions

    Quote Originally Posted by TDA120 View Post
    But wonít I be using integrals already then by using the log rules;
    lnpq -lnp=lnp-ln1
    lnp+lnq-lnp=lnp-0
    lnq=lnp
    Because this is exactly what I shouldnít be doing...
    You are exactly right about that. But that property, \log(pq)=\log(p)+\log(q), is usually proven using the integral definition of logarithm: as p>0 then \log(p)=\int_1^p {\frac{{dx}}{x}}.
    What confuses me is that this is not strictly a geometric proof.
    Can you use a analytic proof?
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    Re: Proof of Mercatorís deduction of logarithms from a hyperbola functions

    No this is exactly what I thought the problem was, but it confuses me. The teacher couldnít solve it himself, he was struggling with ratioís then said: íI know itís a simple trick but I canít remember it right now' and then boom it was homework
    The whole class was about logs in history and Iím afraid the point he wanted to make was something else than a summation of rectangles. But unfortunately I donít have it clear myself.

    (And oops I made a mistake
    lnpq -lnq=lnp-ln1
    lnp+lnq-lnq=lnp-0
    lnp=lnp
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    Re: Proof of Mercatorís deduction of logarithms from a hyperbola functions

    Quote Originally Posted by TDA120 View Post
    No this is exactly what I thought the problem was, but it confuses me. The teacher couldnít solve it himself, he was struggling with ratioís then said: íI know itís a simple trick but I canít remember it right now' and then boom it was homework
    The whole class was about logs in history and Iím afraid the point he wanted to make was something else than a summation of rectangles. But unfortunately I donít have it clear myself.
    I learned this from Leonard Gillman. His textbook defines an integral in terms of area.
    The he defines p>0,~\log(p)=\int_1^p {\frac{{dx}}{x}}
    Using a substitution, x = pt, we note that \int_p^{pq} {\frac{{dx}}{x}}  = \int_1^q {\frac{{dt}}{t}}

    So \log(pq)=\int_1^{pq} {\frac{{dx}}{x}}  = \int_1^p {\frac{{dx}}{x}}  + \int_p^{pq} {\frac{{dx}}{x}}  = \int_1^p {\frac{{dx}}{x}}  + \int_1^q {\frac{{dt}}{t} = \log (p) + \log (q)}
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    Re: Proof of Mercatorís deduction of logarithms from a hyperbola functions

    Hi all,
    I am working with TDA120 on this question which is due in about 24 hours, so quick responses are most valuable!

    The problem is to solve this without using integrals, Instead we must prove this geometrically, so with rectangles, ratios etc.

    Here again the original homework question:
    Prove geometrically that the area under the hyperbola (y=1/x) from 1 to p, is equal to the area from q to pq. Deduce that the area (as function) has the addition property of logarithms, log(ab)=log(a)+log(b).

    Cheers for any thoughts on this! x
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    Super Member ILikeSerena's Avatar
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    Re: Proof of Mercatorís deduction of logarithms from a hyperbola functions

    The region can be stretched horizontally, and reduced vertically, which gives a region with the same surface area.

    To verify that the curve is the same, any point in between can be checked to give the proper height.

    (This is similar to the geometric interpretation of an integral.)
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