Shouldn't 'antilogarithm' be called 'logarithm'?

Question

$$\log_ab=c$$

1. logos = ratio, referring to a;

2. arithmos = number, referring to b;

   therefore

3. logarithm = number (arithmos, b) to be expressed in terms of a ratio (logos, a).

So the logarithm itself is b, while c is the index to the logarithm generated from the logos/base a.

However it's common to speak as if c is the logarithm (as in "the logarithm of 100 to base 10 is 2") and b the antilogarithm. Isn't that confusing?

Answer 1

You got the references all mixed up. The logos refers to $c,$ and the arithmos refers to $b.$

Answer 2

$$\log_ab=c$$

I'm now of the view that the normal way of speaking ("the logarithm of 100 to base 10 is 2") is correct, with the logarithm itself being c, the 'ratio-number' which indexes the position of b in the geometric sequence whose ratio (and, since Briggs, also base) is a.

Others suggest that "reckoning number" is the true derivation. I think the two derivations are compatible, and I find the "ratio-number" derivation more useful for understanding the past and present function. If anyone can quote Napier on this point, I'd be glad to hear it.

Thanks to all who answered and commented.

Answer 3

Long comment

See Brian Rice & Enrique González-Velasco & Alexander Corrigan, The Life and Works of John Napier (2017, Springer), page 404-5, for the original context of Napier's discovery (please, note the kinematic model) :

Let the line $TS$ be radius, and $dS$ a given [arbitrary] sine in the same line: let $g$ move geometrically from $T$ to $d$ in certain determinate moments of time.[...]

Thus if the line $bi$ is as described above, if $dS$ is an arbitrary sine —possibly a whole number chosen from a published table— if the points $g$ and $a$ start their motions with identical initial velocities, and if $a$ arrives at $c$ in the same amount of time as $g$ arrives at $d$, the distance $bc$ —not necessarily an integer in this general case— represents what we could call the generating exponent of the sine $dS$.

Actually, Napier did not use the word logarithm in the Constructio. He called the distance $bc$ the “artificial number” of the sine $dS$. It must have been later, when he was ready to publish the Descriptio, that he introduced the word now in use. He may have realised that, for instance, $4S$ is obtained from $TS$ after four multiplications by the ratio $0.9999999$, in the case of the first table. Thus, the artificial number of $4S$ is $4$, the number of ratios needed to obtain this sine from the whole sine. Using the Greek words logon = ratio and arithmos = number, he concocted the word logarithm or number of ratios. This is the prevailing theory, but it has also been argued that Napier took the word logarithm directly from the The Sand-Reckoner of Archimedes. Regardless of its origins, “logarithm” was the word used instead of “artificial number” in the first translation of the Constructio.