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For exponential functions in calculus, is using 'e' as often as possible, best practice?

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It depends on context. If the **a** is important to what you are describing then use it, if the base of the exponent is not important than we usually default to e.
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i dont think it makes much of a difference i would keep it as Ca\^x but you could totally switch it to Ce\^ln(a) if you wanted to.
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The reason to write C e^(x ln a) is because when you go to differentiate it or integrate it, you don't need to remember a formula for for exponential expressions with other bases.

d/dx (e^(x ln a)) = ln a e^(x ln a) = ln a a^x

So you can either turn a^x into e^(x ln a) or you can remember d/dx (a^x) = ln a a^x as a separate "rule" of differentiation.

If it were me, I'd err on the side of memorizing fewer things.

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