What is an exponential function?

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Hacp

Lifer
Jun 8, 2005
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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?
 
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TecHNooB

Diamond Member
Sep 10, 2005
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A function which grows exponentially is an exponential function! Just plot the damn things and look at it. Does it have linear or exponential behavior? I think you're looking too closely at the variables and notations and whatnot for an answer. At the end of the day, if you plot it, and it exponentially grows or decays, it's an exponential function.
 
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Fenixgoon

Lifer
Jun 30, 2003
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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. growth increases with the power (exponent) of the base. so e^x for x = 1 is just e. e^x when x=5 is e^5. this is not 4x greater than e but 4 *factors* of e greater. 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.
 
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Nik

Lifer
Jun 5, 2006
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Hacp

Lifer
Jun 8, 2005
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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.

What about e^(x^100/5^x). What kind of function is that?
 

TecHNooB

Diamond Member
Sep 10, 2005
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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!
 

artikk

Diamond Member
Dec 24, 2004
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The base doesn't have to be e for it to be an exponential function either. It could be 2^x or whatever rational number you want as the base.
 

Cogman

Lifer
Sep 19, 2000
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What about e^(x^100/5^x). What kind of function is that?

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 :)
 
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Hacp

Lifer
Jun 8, 2005
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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!

That is not my question. If the function is e^x or 2^x or whatever and you set X=something like siny or cosy or (100+y!)/y^y is it still considered an exponential function? What are the rules that govern what is exponential and what isn't?
 

Hacp

Lifer
Jun 8, 2005
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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?
 

Cogman

Lifer
Sep 19, 2000
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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.
 

Hacp

Lifer
Jun 8, 2005
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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.

Ok thank you. That answered my question.
 
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