Powers of trigonometric functions

When writing a power of a trigonometric function or logarithm, it's common to write the power as a superscript before the brackets:

\[ \sin^2(x) = (\sin(x))^2 \]

This conflicts with the conventional notation for inverses and repeated application of functions:

\[ f^2(x) = f(f(x)) \]

\[ y = f^{-1}(x) \implies x = f(y) \]

In his entry on Notation in the Edinburgh Encyclopedia, Charles Babbage is scathing about this convention.

"Although a definition can not be false, it may be improper."