How Were Napier’s Logarithms Invented?

[DLeuPel]

I’m trying to figure out how logarithms we’re invented. In addition, what does the calculator do when I want to solve a logarithm. After researching I found out that you could compare an arithmetic progression with a geometrical one, obtaining the principal properties of exponent calculation. Later, I found that Napier imagined a segment AB, where there was a point Q traveling along the segment but it’s velocity decreased proportionally in relation to the distance left to reach B. This would be a geometric progression. Then, you would divide the segment for each second that passed. After that, you would stretch the divided parts so they are equal. So due to the fact that the velocity decreases in relation to the distance left to B, it never reaches B. Therefore, the distance to B is infinite since for each second, the point travels the same distance. This would be an arithmetical progression.

Until there I can sort of understand it, but the difficult part for me comes here:

—————- The point

moves at a constant speed of 10⁷, so we have

Since P moves at a speed that is proportional to the distance

left to travel we have

From this we see that

which gives

for some constant

We can work out the value of c using our initial conditions. At the start, the point P still needs to travel the whole length of the line segment AB, which is 10⁷. Therefore

. The point Q hasn’t gone anywhere yet, so

Plugging this into the expression above gives

so

Therefore,

so

and

————-

All of the calculations on the top are copied from https://plus.maths.org/content/calculating-napiers-logarithm

Now, if we are trying to solve or to figure out how were logarithms invented and imagined, how does the ln appear if that itself is a logarithm. Also, could someone complete the explication of the foundation of logarithms ?

 

[Svein]

Now, if we are trying to solve or to figure out how were logarithms invented and imagined, how does the ln appear if that itself is a logarithm. Also, could someone complete the explication of the foundation of logarithms ?

Well, multiplication used to be hard and addition easy (by comparison). Therefore, the invention of logarithms meant that you could look up the factors in a table, add them, and then look up the answer in (another) table.

And - it laid the groundwork for the invention of the "slide rule", the essential part of any engineer's tool set.

 

[DLeuPel]

Well, multiplication used to be hard and addition easy (by comparison). Therefore, the invention of logarithms meant that you could look up the factors in a table, add them, and then look up the answer in (another) table.

And - it laid the groundwork for the invention of the "slide rule", the essential part of any engineer's tool set.

I was thinking more of a mathematical approach for an explanation, but thank you.

 

[Svein]

I was thinking more of a mathematical approach for an explanation, but thank you.

Mathematical approach - check out https://en.wikipedia.org/wiki/History_of_logarithms.