e not equal to exp in Maple 2019.1

Question

Can anyone explain why Maple 2019.1 does not recognize that e and exp are equivalent?

Answer 1

The answer to your original question is that e is just a letter in 1D plaintext Maple notation, and doesn't mean anything. In particular it is not related to the base of the natural logarithm, and it is not equal to exp(1). They are not equivalent syntax.

As for your followup comment, ln(exp(x)) is not equal to x for all complex values of x, which is why Maple does not simplify the former to the latter. In this context Maple will consider an unknown name such as x to be complex-valued, and possibly nonreal.

Look at the help page for topic ln, which explains in the second paragraph of its description,

For complex-valued expressions x, 
          ln(x) = ln(abs(x)) + argument(x)*I
where
          -Pi <argument(x) <= Pi
Throughout Maple, this computation is taken to be the
definition of the principal branch of the logarithm.

Perhaps you were thinking of purely real x? If so then you have to inform Maple of that special aspect -- it doesn't read your mind.

simplify(ln(exp(x))) assuming x::real;
                           x

Some more examples,

simplify(ln(exp(A+B*I))) assuming A::real,B::real,B>0,B<Pi;

                        A + I B

simplify(ln(exp(1+3/2*Pi*I)));

                           1     
                       1 - - I Pi
                           2     

simplify(evalc(ln(exp(A+B*I))));

              A + I arctan(sin(B), cos(B))

And, plotting part of that last expression,

plot( arctan(sin(B), cos(B)),
      B = -3.5*Pi..3.5*Pi,
      xtickmarks=piticks );