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http://mathworld.wolfram.com/images/eps-gif/LogarithmicallyDecreasingFunction_1000.gif

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?

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    [Wolfram MathWorld](http://mathworld.wolfram.com/LogarithmicallyDecreasingFunction.html) backs you up. However, [exponential decay](https://mathbitsnotebook.com/Algebra1/FunctionGraphs/expdecay2.jpg) does look pretty similar. – 2017-02-27
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    That post describes the graph as a "*Logarithmically Decreasing Function*" - contrarily. – 2017-02-27
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    Read the definitions at MathWorld. Logarithmically decreasing means decreasing slower than any polynomial, while exponentially decreasing means decreasing faster than any polynomial. I'm not sure how that's determined for a given function, though. – 2017-02-28
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    I don't really get what that means. Is it that, a logarithmically decreasing graph will drop off, at a slower rate than any $x^k$ graph? – 2017-02-28
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    I believe it means that $f$ is logarithmically decreasing iff $$f(x) > 0 \text{ for all } x\qquad \text{and}\qquad \lim_{x\to \infty} f(x) = 0\qquad \text{and}\qquad \lim_{x\to \infty} f(x)p(x) = \pm \infty$$ for any nonconstant polynomial $p$. – 2017-02-28

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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.

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    I was asking in regards to exponential decay equations in my physics syllabus, of the form: $y=ke^{ax}$, but assumed this graph was similar to the one I included in the question. Its good to know, though, that there's a distinct difference. – 2017-02-28