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Thread: Why logarithmic functions instead of radical functions?

  1. #1
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    What is the relationship between logarithms and radicals?

    Here's my question: what is the relationship between radicals and logarithms? How are they related?

    (My original question is below, but the answer occurred to me after I posted it)

    I have a simple question--why is the inverse of an exponential function a logarithmic function instead of a radical function?

    If $\displaystyle F(x) = 5^x$, wouldn't the inverse be $\displaystyle x = 5^y$, and couldn't that be written $\displaystyle \sqrt[5]{x}$? Why must it be written $\displaystyle log_5x$
    Last edited by shirkdeio; Dec 4th 2008 at 07:43 AM. Reason: new question
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  2. #2
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    Quote Originally Posted by shirkdeio View Post
    Here's my question: what is the relationship between radicals and logarithms? How are they related?

    (My original question is below, but the answer occurred to me after I posted it)

    I have a simple question--why is the inverse of an exponential function a logarithmic function instead of a radical function?

    If $\displaystyle F(x) = 5^x$, wouldn't the inverse be $\displaystyle x = 5^y$, and couldn't that be written $\displaystyle \sqrt[5]{x}$? Why must it be written $\displaystyle log_5x$
    Hi

    If $\displaystyle y = 5^x$ then $\displaystyle x = log_5 \,y$

    If $\displaystyle y = x^5$ then $\displaystyle x = \sqrt[5] y$
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  3. #3
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    Quote Originally Posted by running-gag View Post
    Hi

    If $\displaystyle y = 5^x$ then $\displaystyle x = log_5 \,y$

    If $\displaystyle y = x^5$ then $\displaystyle x = \sqrt[5] y$
    Thanks. I don't know why I had trouble with that--logarithms confuse me.

    But is $\displaystyle y = 5^x$ the inverse of $\displaystyle \sqrt[x]{y} = 5$? If so, what is the relationship/difference between the radical and $\displaystyle y = log_5{x}$?
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  4. #4
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    Quote Originally Posted by shirkdeio View Post
    Thanks. I don't know why I had trouble with that--logarithms confuse me.

    But is $\displaystyle y = 5^x$ the inverse of $\displaystyle \sqrt[x]{y} = 5$? If so, what is the relationship/difference between the radical and $\displaystyle y = log_5{x}$?
    The inverse function of y=f(x) is $\displaystyle x = f^{-1}(y)$
    You have to express x function of y

    If $\displaystyle y = 5^x$ then $\displaystyle x = log_5\,y$

    $\displaystyle \sqrt[x]{y} = 5$ is also true but it is not in the form $\displaystyle x = f^{-1}(y)$

    It is the same as per the example :
    If $\displaystyle y = 5x$ then $\displaystyle x = \frac{y}{5}$

    $\displaystyle \frac{y}{x} = 5$ is also true but it is not in the form $\displaystyle x = f^{-1}(y)$
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  5. #5
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    Thank you. I think I better understand logarithms now. I appreciate the help!
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