Proof of Mercator’s 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 need’s to be done geometrically; something to do with ratio’s.

Of course the area of a rectangle formed by op*pp’ (let p’ be 1/p) equals oq*qq’ and opq*pq(pq)’

But I’m not getting any further staring at my hyperbola...

Re: Proof of Mercator’s 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...

Re: Proof of Mercator’s deduction of logarithms from a hyperbola functions

But won’t 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 shouldn’t be doing...

Re: Proof of Mercator’s 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.

Re: Proof of Mercator’s 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!

Re: Proof of Mercator’s deduction of logarithms from a hyperbola functions

Quote:

Originally Posted by

**TDA120** **But won’t 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 shouldn’t be doing...

You are exactly right about **that**. But that property, $\displaystyle \log(pq)=\log(p)+\log(q)$, is usually proven using the integral definition of **logarithm**: as $\displaystyle p>0$ then $\displaystyle \log(p)=\int_1^p {\frac{{dx}}{x}}$.

What confuses me is that this is **not strictly a geometric proo**f.

Can you use a analytic proof?

Re: Proof of Mercator’s deduction of logarithms from a hyperbola functions

No this is exactly what I thought the problem was, but it confuses me. The teacher couldn’t solve it himself, he was struggling with ratio’s then said: ’I know it’s a simple trick but I can’t remember it right now' and then boom it was homework :(

The whole class was about logs in history and I’m afraid the point he wanted to make was something else than a summation of rectangles. But unfortunately I don’t have it clear myself.

(And oops I made a mistake

lnpq -lnq=lnp-ln1

lnp+lnq-lnq=lnp-0

lnp=lnp

Re: Proof of Mercator’s deduction of logarithms from a hyperbola functions

Quote:

Originally Posted by

**TDA120** No this is exactly what I thought the problem was, but it confuses me. The teacher couldn’t solve it himself, he was struggling with ratio’s then said: ’I know it’s a simple trick but I can’t remember it right now' and then boom it was homework :(

The whole class was about logs in history and I’m afraid the point he wanted to make was something else than a summation of rectangles. But unfortunately I don’t have it clear myself.

I learned this from Leonard Gillman. His textbook defines an **integral**** in terms of **__area__.

The he defines $\displaystyle p>0,~\log(p)=\int_1^p {\frac{{dx}}{x}}$

Using a substitution, $\displaystyle x = pt$, we note that $\displaystyle \int_p^{pq} {\frac{{dx}}{x}} = \int_1^q {\frac{{dt}}{t}}$

So $\displaystyle \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)} $

Re: Proof of Mercator’s 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

Re: Proof of Mercator’s 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.)