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    I get that they're reverse operations but am still really confused as to why the answer is 3.

    Is there a method so I could see how exactly they cancel to get 3.


    Posted from TSR Mobile
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    why does 2 + 1 - 1 = 2?

    because maths
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    (Original post by Excuse Me!)
    I get that they're reverse operations but am still really confused as to why the answer is 3.

    Is there a method so I could see how exactly they cancel to get 3.


    Posted from TSR Mobile
    It's basically the same reason as to why \sqrt{a^2} = a, or \sin(sin^{-1}(x)) = x.

    If you said f(x) = e^x, then we have f^{-1}(x)=\ln x.

    Then we know that f(f^{-1}(x)) = x (by definition of them being inverses to each other), and so f(f^{-1}(x)) = f(\ln x) = e^{\ln x} = x.
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    (Original post by Excuse Me!)
    I get that they're reverse operations but am still really confused as to why the answer is 3.

    Is there a method so I could see how exactly they cancel to get 3.


    Posted from TSR Mobile
    Er, you don't "get" it because if you did you would understand why it worked that way

    I'm not being funny, but the definition of inverse functions is that they 'undo' each other's actions and bring back the original number.


    So, take the number 3, apply ln to it to produce another number ln 3. We know that e^x is the inverse of the function ln x, so if we exponentiate ln 3 we get back our original number i.e 3 - in other words e^{ln 3} = 3

    But we could also do this:

    Start with 3, exponentiate it to get e^3 which is just another number. We know that ln x is the inverse of the function e^x so if we apply the ln function to our new number e^3 we get back our original number 3 i.e.

    ln(e^3) = 3

    Does this make things any clearer?
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    (Original post by Excuse Me!)
    I get that they're reverse operations but am still really confused as to why the answer is 3.

    Is there a method so I could see how exactly they cancel to get 3.


    Posted from TSR Mobile
    Think of ln(x) as asking the question "e to the power of what is x"

    So if x = e^3 , then clearly "e to the power of 3 is x", so ln(e^3) = 3.
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    (Original post by brittanna)
    It's basically the same reason as to why \sqrt{a^2} = a, or \sin(sin^{-1}(x)) = x.

    If you said f(x) = e^x, then we have f^{-1}(x)=\ln x.

    Then we know that f(f^{-1}(x)) = x (by definition of them being inverses to each other), and so f(f^{-1}(x)) = f(\ln x) = e^{\ln x} = x.
    This statement is wrong sqrt(a^2)=|a| for real a not a.
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    ln3 basically says what number do we raise e by to get 3
    So e^ln3 is raising e to that power - to the power that we raise e by to get 3. So we get 3
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    Well first of all this question wouldn't make sense without knowing the definition of the function f(x)=ln(x) just like 1+1 doesn't make sense without knowing what + means. Therefore we need to define ln(x). This is defined as the inverse of g(x)=e^x. So by definition f(x) satisfies g(f(x))=x, i.e. e^(f(x))=x, which is what the question wants since f(x)=ln(x) (then we just put x=3).

    So this question follows from the definition of f(x).

    In fact, all of maths follows from the definitions, just like you only need the definitions of geometry and the obvious axioms (which mostly can be taken for granted without stating, because they're obvious). But the fun thing about maths is that there are also deductions; this isn't available in a subject such as languages. In languages, a beginning student must study very simple exercises that bear no resemblance to the actual subject of languages, with its beauty, depth and elegance. He must study trivialities such as: convert "the dog barked" into latin, and similar rubbish. Whereas on the other hand, a mathematics student, even a beginner, can be given exercises early on that are as close in originality and spirit as the greatest theorems of his subject. Such are for example many olympiad problems in geometry, many of which I am sure even the ancient greek geometers would have been proud of.

    This is why mathematics is such a beautiful subject. Indeed, the whole subject stems from deductions from the axioms/definitions, just like a tree stems from the seed and its roots. Edit: This is also why it has the greatest proportion of cranks (physics is second), because they think they are doing original work by discovering deductions, but more often than not their "deductions" are simply restatements of a fundamental principle or well-known theorem (while the rest usually fall into the category of "olympiad problem").
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    (Original post by NEWT0N)
    Well first of all this question wouldn't make sense without knowing the definition of the function f(x)=ln(x) just like 1+1 doesn't make sense without knowing what + means. Therefore we need to define ln(x). This is defined as the inverse of g(x)=e^x. So by definition f(x) satisfies g(f(x))=x, i.e. e^(f(x))=x, which is what the question wants since f(x)=ln(x) (then we just put x=3).

    So this question follows from the definition of f(x).

    In fact, all of maths follows from the definitions, just like you only need the definitions of geometry and the obvious axioms (which mostly can be taken for granted without stating, because they're obvious). But the fun thing about maths is that there are also deductions; this isn't available in a subject such as languages. In languages, a beginning student must study very simple exercises that bear no resemblance to the actual subject of languages, with its beauty, depth and elegance. He must study trivialities such as: convert "the dog barked" into latin, and similar rubbish. Whereas on the other hand, a mathematics student, even a beginner, can be given exercises early on that are as close in originality and spirit as the greatest theorems of his subject. Such are for example many olympiad problems in geometry, many of which I am sure even the ancient greek geometers would have been proud of.

    This is why mathematics is such a beautiful subject. Indeed, the whole subject stems from deductions from the axioms/definitions, just like a tree stems from the seed and its roots. Edit: This is also why it has the greatest proportion of cranks (physics is second), because they think they are doing original work by discovering deductions, but more often than not their "deductions" are simply restatements of a fundamental principle or well-known theorem (while the rest usually fall into the category of "olympiad problem").
    +rep for effort and motivation which I lack these days
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    (Original post by B_9710)
    ln3 basically says what number do we raise e by to get 3
    So e^ln3 is raising e to that power - to the power that we raise e by to get 3. So we get 3
    Actually a good response rather than the smug stupid answers other people give.
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    (Original post by Rad-Reloaded)
    Actually a good response rather than the smug stupid answers other people give.
    from a teaching point of view I agree.
    from a mathematicians point of view this is an incorrect response.
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    (Original post by TeeEm)
    from a teaching point of view I agree.
    from a mathematicians point of view this is an incorrect response.
    There is nothing mathematically incorrect about what I said. It may not be as technical as a 'mathmetician' would like but what I said can help people to understand it.
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    (Original post by B_9710)
    There is nothing mathematically incorrect about what I said. It may not be as technical as a 'mathmetician' would like but what I said can help people to understand it.
    Isn't that exactly what I said?
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    (Original post by TeeEm)
    Isn't that exactly what I said?
    No you said it was an incorrect response from a mathmetician a point of view. It's not incorrect from any point of view - it can't be because it is not incorrect
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    (Original post by B_9710)
    No you said it was an incorrect response from a mathmetician a point of view. It's not incorrect from any point of view - it can't be because it is not incorrect
    I am not going to engage into a debate or an argument with you ...
    I sincerely wish you all the best
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    (Original post by brittanna)
    It's basically the same reason as to why \sqrt{a^2} = a
    This is incorrect.

    \sqrt{a^2} = |a|
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    (Original post by TeeEm)
    I am not going to engage into a debate or an argument with you ...
    I sincerely wish you all the best
    k.
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    (Original post by ubisoft)
    This is incorrect.

    \sqrt{a^2} = |a|
    I am sure brittanna knows that, but I am sure he did that for the sake of simplicity, given the OP did not understand why e^ln3 = 3
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    (Original post by TeeEm)
    I am sure brittanna knows that, but I am sure he did that for the sake of simplicity, given the OP did not understand why e^ln3 = 3
    Ah fair enough. Can you explain how what the other user said about ln is incorrect? That's how I still think about it too, would like to know if it's wrong.
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    (Original post by TeeEm)
    I am not going to engage into a debate or an argument with you ...
    I sincerely wish you all the best
    (Original post by B_9710)
    No you said it was an incorrect response from a mathmetician a point of view. It's not incorrect from any point of view - it can't be because it is not incorrect
    who even knew drama could arise in the maths forum...about maths :lol:
 
 
 
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