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How does one convert a natural exponent written as $e^{c\cdot x}$ into the form $a^{x}$ ?

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    @MichaelHardy Well put. I concede.2012-03-22

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Since you can put $e^c=a$, you have that

$e^{c \cdot x} = (e^c)^x=a^x$

If you have that $a$ is some other number you can choose $c=\log a$ to get $e^c = a$