Why is it called "antilog" or "anti-logarithm" rather than exponentiation?

I always get a little annoyed when engineers or math textbooks use the word "antilogarithm." Isn't it just exponentiation? Like if $\log(2) \approx 0.301$, then $10^{0.301}\approx 2$ . Why say "antilogarithm?" Is there some subtly different meaning? Am I missing something?

Solution 1:

$\DeclareMathOperator\antilog{antilog}$

First, when Napier invented logarithms, his application was not the inverse of the exponential function. He was originally multiplying quantities called "sines", which are not what you're thinking when you see that word. These "sines" are vaguely similar to the positions of the arrow in Zeno's paradox of the arrow. To each "sine" (a quantity) was associated a quantity, the logarithm of that "sine". Napier arranged for his logarithms to have the property that when the "sines" decreased in geometric proportion, the logarithms increased in arithmetic proportion. So he transformed multiplication of "sines" into addition of their logarithms.

Now to your question: to go from a "sine" to its logarithm was performed by table lookup. To go from a logarithm to its "sine" was performed by backwards table lookup. So, quite literally, an antilogarithm is found by using the table backwards. Since we are not working on usual quantities, but on "sines", the inverse operation is not exponentiation.

Napier and Briggs then modified the logarithm to work with "normal" quantities instead of "sines". At this point, the inverse of the logarithm was exponentiation.

Note that the inverse of the logarithm couldn't be called exponentiation by Napier since he was writing in 1614 and subsequently. In 1748 Euler wrote "consider exponentials or powers in which the exponent itself is a variable" ("Primum ergo considerandæ sunt quantitates exponentiales, seu Potestates, quarum Exponens ipse est quantitas variabilis.", from Introductio in analysin infinitorum) , which seems to be the first time the exponent was not a constant positive integer. Until we make the generalization of exponents to arbitrary powers, there is no hope of describing the inverse logarithm as an exponential function.

One "convenience" of the antilog notation is that the following equation $$ \log \antilog x = x = \antilog \log x $$ is true both for Napier's "sines" and subsequent inverse exponential logarithms. Rewriting this where the base is variable (which is not what Napier was considering) $$ \log_b \antilog_b x = x = \antilog_b \log_b x \text{,} $$ which is (as many students have shown) more parsable than $$ \log_b b^x = x = b^{\log_b x} \text{.} $$