A good reference is John Napier and the invention of logarithms, 1614, by E. W. Hobson (1914). You can also read the original source works by Napier: Mirifici logarithmorum canonis descriptio (English: A description of the admirable table of logarithmes, 1616), and Mirifici logarithmorum canonis constructio (English: The construction of the wonderful canon of logarithms, 1889).
Here is a summary.
We now think of the logarithm function as the inverse of exponentation, but Napier was working at a time when even the notion of exponentiation was not a common one. (This was all also before calculus, computation with infinite series, or coordinate geometry.) Instead, his crucial insight was a certain definition of "logarithms" which satisfied the following
Proposition: If a set of numbers is in geometric progression, then their logarithms are in arithmetic progression.
So, roughly speaking, Napier constructed sets of numbers in geometric progression, and found their logarithms using linear interpolation. For the geometric progressions, he chose common ratios like $(1-\frac1{10^7})$ (others he used were $(1-\frac1{10^5})$, $(1-\frac1{2000})$ and $(1-\frac1{100})$) because it's easy to multiply a number by it: subtract from the number the result of shifting it $7$ decimal places to the right. For instance, to seven decimal places, $9999998.0000001(1-\frac1{10^7})$ is
$$\begin{align} 9999998&.0000001 \\ -\phantom{000000}0&.9999998 \\ -----&----- \\ = 9999997&.0000003\end{align}$$
This is what Wikipedia means by "by repeated subtraction".
Specifically, his logarithm was defined as follows.
Imagine a line $TS$ of length $R$ (= $10^7$), along which a point $P$ moves from $T$ to $S$ such that its velocity is proportional to its distance from $S$. Meanwhile, another point $Q$ on a different line, starting at $T_1$ when $P$ is at $T$, moves at uniform velocity (the velocity of $P$ when at $T$).
DEFINITION: If when the point $P$ is at $P_1$ the point $Q$ is at $Q_1$, then the logarithm of the length $P_1S$ is defined to be the length $T_1Q_1$.
So $l(R)=0$ and $l(x) \to \infty$ as $x \to 0^+$. Also $l(x)<0$ for $x>R$.
Napier showed, with an essentially sound argument, that when $Q_1, Q_2, \dots$ are covered after equal times, i.e., the lengths $T_1Q_1, T_1Q_2, \dots$ are in arithmetic progression, then the lengths $P_1S, P_2S, \dots$ are in geometric progression. This is the proposition mentioned above.
In modern notation, the Napier logarithm $l(x)$ he defined is (and what he calculated is an approximation to) $10^7 \log_{ \frac{1}{e}} \left(\frac{x}{10^7} \right) = -10^7\ln x + 10^7\ln (10^7)$. We can see this as follows: if $x$ is the length $PS$, and $y = l(x)$ is the length $T_1Q$, then $\frac{dx}{dt} = - \frac{Vx}{R}$ where $V$ is the initial velocity when $P$ is at $T$, and $\frac{dy}{dt} = V$, so $\frac{dy}{dx} = - \frac{R}{x}$, and solving this differential equation (with the initial condition $l(R)=0$) gives $y = -R\ln x + R\ln R$. (Involved here is the observation that $\frac{d}{dx} \ln x = \frac1x$, which seems due to Newton a few decades later. See the timeline at the bottom of this page.)
The first few values are more or less equal to $\log_{(1-10^{-7})} \left( \frac{x}{10^7} \right)$. (To see this approximately, like Napier: pick some small time unit, and suppose that after $1$ time unit $y=k$ and $x = Rr$ for some $r<1$. (If the time unit is small, the distance $R(1-r) \approx k$.) Then after $t$ time units $y = kt$ and $x = Rr^t$. So $t = \log_r(x/R)$ and $y = k\log_r(x/R)$ which for $r = (1-1/R)$ and $k \approx R(1-r) = 1$ gives $y = \log_{1-1/R}(x/R)$.) Later parts of his table are approximations in different ways, depending on the corresponding geometric progressions.
The modern definition of logarithms so that $\log 1 = 0$ and $\log 10 = 1$ (well, he was thinking of $\log 10 = 10^{10}$, but this just means computing logarithms to ten digits) occurred to Napier only after he had begun working on his original plan, so in his publications he only proposes an outline of how this table could be constructed. It was Henry Briggs who published the first table of base-10 logarithms.
To be precise, Napier's table gave the "logarithms" of sines of angles from $0^\circ$ to $90^\circ$. The then definition of $\mathop{Sine} \theta$, dating all the way back from Aryabhata in the 5th century, was (for some fixed radius $R$) the length of the half-chord that subtends angle $\theta$ in a circle of radius $R$. In modern notation, $\mathop{Sine} \theta = R\sin \theta$. Napier chose $R = 10^7$. (So $\mathop{Sine} 0^\circ = 0$ and $\mathop{Sine} 90^\circ = 10^7$.) His table gave the "logarithms" of the Sines of equidistant angles, so although it gave logarithms of numbers from $0$ to $10^7$, these numbers were not equally spaced. Here is part of a page from his tables (see more here):
The last row says (in modern notation, and omitting digits after the decimal point) that $10^7\sin(9^\circ 15') = 1607426$, that $l(1607426) = 18279507$ (actually it should have been $18279511$), and reading from the right, it gives the Sine of the complement angle $80^\circ 45'$ (or equivalently the Cosine of that angle), the logarithm of that Sine, and the difference between the two logarithms (useful essentially because $\log\sin\theta -\log\cos\theta=\log\tan\theta$, etc).
About the choice of $10^7$ (concisely explained in Henry's answer), note that at the time decimal notation was also not standard. So instead of putting digits after the decimal point and expecting the readers to understand, he had to multiply by a large power of 10 so that enough digits are in the integral part (before where the decimal point would be). Hobson's book says
His choice was made with a view to making the logarithms of the sines of angles between $0^\circ$ and $90^\circ$, i.e., of numbers between $0$ and $10^7$, positive and so as to contain a considerable integral part.
And Napier's Constructio says
In these progressions we require accuracy and ease in working. Accuracy is obtained by taking large numbers for a basis; but large numbers are most easily made from small by adding cyphers [zeroes]. Thus instead of 100000, which the less experienced make the greatest sine, the more learned put 10000000, …
And so on. It's a bit humorous really, from our perspective, but the Constructio is illuminating for historians who want to know how the first logarithm tables were constructed.
Logarithms were not motivated by the desire to get exact solutions to problems such as multiplying two numbers with $15$ significant digits each.
With a decent table of logarithms you might have a precision of five digits, so indeed the answer you would get by using logarithms to compute
$101{,}123{,}958{,}959{,}055 \times 342{,}234{,}234{,}234{,}236$
would be no more accurate than if you simply multiplied
$101{,}120{,}000{,}000{,}000 \times 342{,}230{,}000{,}000{,}000.$
It might be possible to go to more digits with a better table and sophisticated interpolation techniques, but not many more digits.
Adding two five digit numbers is still easier to do than multiplying two five-digit numbers if the only tools at your disposal are pencil and paper.
This still might not have been enough motivation to purchase a table of logarithms if you only needed to do such a multiplication once a year. But there are problems that require you to perform many multiplications to get just one answer.
Therefore there was an incentive to develop and publish tables of logarithms and promote their use.
Meanwhile, logarithms also turned out to have some interesting mathematical properties, for example providing a solution for the integral of $\frac1x$
(the only power of $x$ whose integral is not another power of $x$).
I think the main reason we study logarithms today is because of things like that, not because we want to get the product of two $n$-digit numbers.
Best Answer
Thanks for bringing this to my attention. I checked the lecture tape, and found that what Feynman originally said was, "When I took a small fraction of 1024, as the fraction went to zero, what would the answer be here?" This was edited in 1961 by Robert Leighton into the form it now takes in The Feynman Lectures on Physics. I agree with you that the current version is a bit confusing, and will recommend that it be changed from, "When we take a small fraction Δ of 1024 as Δ approaches zero..." to, "When we take a small fraction Δ/1024 as Δ approaches zero..."
Mike Gottlieb [Editor, The Feynman Lectures on Physics New Millennium Edition]