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  1. #1 expanding 
    Forum Ph.D. Heinsbergrelatz's Avatar
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    hello to all science forum members,
    recently i have constantly been coming up with the logarithmic power functions in log calculus, and there has been one question that seems impossible to solve but im 100% sure it could be solved without drawing a graph, can anyone help me with this?(its not homework by the way) thank you;



    and also how on earth would you expand with an exponent of something like
    etc.. can anyone give some solution or any idea when this kind of exponent comes out. can you even expand it when it comes to this point? is that why we have the natural logs and all that stuff?


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  3. #2  
    Ots
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    There is an important difference between x^a, where 'a' is some number, and a^x.

    The derivative of x^a is ax^a-1 as you know.

    The derivative of a^x is (a^x)ln a, where 'ln' is the natural logarithm.

    So, yes, there is a section in your calculus book called logarithms and exponential functions and all of the techniques for dealing with variables as exponents are shown.

    Good Reading!


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  4. #3  
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    You do not need to expand an expression such as this.

    But note using the Product Rule for exponents that
    (2x + 1 ) ^ x + 3 = (2x+1)^x times (2x+1)^3
    You can expand the second one easily.

    To solve the equation, I do not think it can be done algebraically
    Either graphically or numerically.
    Note that for x = 1, you are very close to a solution.
    A little trial and error and you will hone in on the solution to whatever
    accuracy you want.
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  5. #4  
    Forum Ph.D. Heinsbergrelatz's Avatar
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    thank you for your responses.
    so, you cant expand in such cases when its the power of
    yes, i was thinking the same, it cannot be solved algebraically though the book says draw a line to the original exponential graph and find an approximate answer to it
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  6. #5 Re: expanding 
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    Quote Originally Posted by Heinsbergrelatz
    hello to all science forum members,
    recently i have constantly been coming up with the logarithmic power functions in log calculus, and there has been one question that seems impossible to solve but im 100% sure it could be solved without drawing a graph, can anyone help me with this?(its not homework by the way) thank you;



    and also how on earth would you expand with an exponent of something like
    etc.. can anyone give some solution or any idea when this kind of exponent comes out. can you even expand it when it comes to this point? is that why we have the natural logs and all that stuff?
    Recall that you can write as .
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  7. #6  
    Forum Radioactive Isotope MagiMaster's Avatar
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    A better rule to try in this case might be . Apply that to the right, then you can take the log of both sides to get rid of the e's.
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  8. #7  
    Forum Ph.D. Heinsbergrelatz's Avatar
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    ooo, ok i see, so you basically just apply log rules,
    alright thank you for the help :-D
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  9. #8  
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    Quote Originally Posted by Heinsbergrelatz
    the book says draw a line to the original exponential graph and find an approximate answer to it
    Yes that is solving it graphically.
    Let y = e^4.5 and graph it.
    Then let y = [the other side of the equation] and graph that.
    Where they intersect are the solutions.

    Graphical solutions are easy to find, especially with a graphing calculator.
    But they are typaically not very precise.
    Thats the tradeoff using that method.
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