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In $b^a = x$, $b$ is the base, a is the exponent and $x$ is the result of the operation. But in its logarithm counterpart, $\log_{b}(x) = a$, $b$ is still the base, and $a$ is now the result. What is $x$ called here? The exponent?

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In $b^a=x$, $x$ (or the whole of $b^a$) is sometimes called the power. – Isaac Sep 7 '10 at 18:35
up vote 18 down vote accepted

Argument (as you call the $x$ in any $f(x)$ argument of the function)

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Another name (that I've only ever seen when someone else asked this question) is "logarithmand".

From page 36 of The Spirit of Mathematical Analysis by Martin Ohm, translated from the German by Alexander John Ellis, 1843:

page from The Spirit of Mathematical Analysis

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Would you mind if I added to your answer an image of the text where "logarithmand" is introduced in the book mentioned in your link? – Jonas Meyer May 10 '11 at 21:12
@Jonas Meyer: Please feel free to add the image; that would be neat to see. Thanks! – Jonas Kibelbek May 10 '11 at 22:20

antilogarithm (noun): a compound of Greek anti-"against" and logarithm (qq. v.). In algebra if $log_{b}(x) = y$, y is the logarithm of x, and x is said to be the antilogarithm of y.

Source: The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, Steven Schwartzman

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I often see things like $\mathrm{antilog}_b\;y$ in engineering and chemistry books, which annoys me a lot because people could have just as easily written $10^y$ or $e^y$ – J. M. Sep 7 '10 at 22:38
@J.M.: True. Much like the weird "conventions" on inverse functions at all. asin, arc, anti, $\bar{f}$, ... If at all, I'd write $log^-1$ – Dario Sep 8 '10 at 20:26
@Dario: I don't even care for the superscript -1 myself. I only use it when showing something to other people but with some gritting of teeth. For personal notes, I use D.J. Jeffrey's hacek notation. – J. M. Sep 8 '10 at 20:57
@Bill: To be upfront, I suppose for me the issue is more aesthetic/psychological than mathematical. I think I'd flip when I see $\exp$ with a subscript. ;) – J. M. Sep 25 '10 at 23:16
@J.M.: The point is uniformity. I'd much rather see $\rm \log_b\ exp_b\ x = x\ $ than $\rm\ \log_b\ antilog_b\ x$. – Bill Dubuque Sep 25 '10 at 23:37

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