What is the $x$ in $\log_b x$ called?

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

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:

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