This isn't really a specific problem, just a question if hand-writing log functions (or trig functions) is interpreted differently than when typing them into a calculator or something like Wolfram Alpha.
Suppose you have this on paper:
Is this the same as both of the expressions below?
This is what you get, when you enter what I think are equivalent expressions to each of those, onto a calculator (or wolfram)
ln (e^x) ===> x
ln (e^1)^x ===> 1
But I guess my question is, is this really the way you'd interpret the above expressions, if you saw them written out?
I thought that if you wanted the ln function taken to a power, you'd write these, for example:
If you do something similar with sine, then wolfram and my TI-82 calculator differ in their interpretations:
.5 (according to wolfram)
.5785... (according to ti-82)
I'm wondering if this is a case where there isn't really one specified standard, or if I'm doing something wrong?
Any insight will be appreciated, thank you!