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I was recently having a discussion with someone, and we found that we could not agree on what an exponential function is, and thus we could not agree on what exponential growth is.

Wikipedia claims it is $e^x$, whereas I thought it was $k^x$, where k could be any unchanging number. For example, when I'm doing Computer Science classes, I would do everything using base 2. Is $2^x$ not an exponential function? The classical example of exponential growth is something that doubles every increment, which is perfectly fulfilled by $2^x$. I'd also thought $10^x$ was a common case of exponential growth, that is, increasing by an order of magnitude each time. Or am I wrong in this, and only things that follow the natural exponential are exponential equations, and thus examples of exponential growth?

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Whether one thinks of exponential growth as $e^{ct}$ ($c$ positive) or $a^t$ ($a\t 1$), one is dealing with the same phenomenon, just take $a=e^c$. –  André Nicolas Jun 21 '12 at 0:45

2 Answers 2

up vote 8 down vote accepted

$e^x$ is the exponential function, but $c\cdot k^x$ is an exponential function for any $k$ ($> 0, \ne1$) and $c$ ($\ne 0$).

The terminology is a bit confusing, but is so well settled that one just has to get used to it.

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Therefore, I can refer to anything that can be described as $c⋅k^x$ as exponential growth, correct? –  Canageek Jun 20 '12 at 23:24
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You can do whatever you want. The question is whether people will understand you, and in this case I think most people will understand. –  Qiaochu Yuan Jun 20 '12 at 23:26
    
The discession was based around the incorrect use of exponential growth (i.e. Hollywood usage) so yeah, I think I'll stick with the proper definitions. Thank you, hope the question wasn't too basic. –  Canageek Jun 21 '12 at 0:49
    
Sometimes, to highlight how special $\exp\,x$ is, it is given the adjective natural, as in "natural exponential function". –  J. M. Jun 21 '12 at 3:20
    
$c\cdot k^x=c \cdot e^{x\cdot \ln k}$, so if constants in the exponent don't bother you then sure. –  Robert Mastragostino Jun 21 '12 at 3:36

If $x$ has units (e.g. time), then there's no way to distinguish between these possibilities; they're all equivalent up to change of units.

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