Would this graph be described as decreasing logarithmically or exponentially (with linear increments of $x$)?
I would think it was logarithmic, as it's decreasing less every time.
My teacher, however, describes this as exponential. Additionally, "exponential decay" follows this graph...
I'm confused, are the terms used some-what interchangeably?
I think you could use either term: $$y=e^{-kx}$$ is the same as $$x=-\frac1k\ln y\ ,$$ so if $y$ is an exponential function of $x$ then $x$ is a logarithmic function of $y$.
HOWEVER, note that this is a very specific type of curve: "exponential" does not just mean "very fast". (The term is often used this way in the media nowadays, but it is not, usually, mathematically correct.) And according to the labels on your graph, the equation appears to be $$y=\frac1{\ln x}\ ,$$ which can be written $$x=e^{1/y}\ .$$ These equations are not the same as above, so this is not exponential/logarithmic growth/decay.