This always confuses me. Is e^lnx exponential? Is e^2x exponential? Is e^-x exponential?I always thought the first one wasn't and the 2nd one was and the 3rd one wasn't, but if that is so, can anyone explain why?
e^ln(x) is linear. e^ln(x) = x. likewise, ln (e^x) = x.
e^x is exponential growth. coefficients multiplying X will either elongate (A*x, A < 1) or contract (A*x, A > 1) the exponential curve. Addition or subtraction - e^(x-5) or e^(x+5) - serve to shift the curve left or right.
e^-x is exponential decay.
e is simply the base number (2.71828...) of the natural logarithm, so e^x is really 2.71828^x.
You sir are an idiot.
What about e^(x^100/5^x). What kind of function is that?
I think what's happened is you've considered 'e^' to be the almighty exponential function, but when you raise it to the ln(x), things fall apart because the result does not have an exponential behavior. Just change your definition and you'll be fine. Exponential behavior is what counts!
It is a complex function, neither exponential nor linear. though, as x->inf, it evaluates to 1.
If you are talking about big O notation, you could actually consider this function to in the constant class of functions! O(1). Isn't that bizarre
If the function takes the form e^(variable) does the variable need to be linear? Or can it be any polynomial? Or can it be any function?
It MUST be a linear function for it to be considered an exponential function, anything else will evaluate to something else.
Another definition is that the derivative must evaluate to the same function varying by some constant value.