Honestly, e belongs in a calculus course, not in algebra 2. Presenting it prematurely makes it seem magical and incomprehensible. Why the rush to get e in before students are ready for it?
The essential question, really, is: what polynomials come closest to satisfying the identity f(x)*f(y) = f(x + y)? One readily passes inductively from the nth-degree best approximation to the (n+1)st-degree best approximation. The answers all contain the same arbitrary constant.
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This constant may be chosen to be 1, which immediately simplifies everything marvelously, and feels like the obvious "best" value. Doing this corresponds to making e the base for the exponentiation operation. That is, it gives access to approximations e, by the values f(1).
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Base 10 logarithms were the norm in the 17th century, but a table of the natural logarithm of all integers from 1 to 1000 was published already in 1622 (Speidell, New Logarithmes), agreeing with the modern ln(x) to six decimal places (though the table omits the decimal point).
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