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Thread: Why is 10^lg 5=5 and e^ln5=5, etc?

  1. #1
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    Why is 10^lg 5=5 and e^ln5=5, etc?

    Why is

    $\displaystyle 10^(lg{5})=5$
    $\displaystyle e^(ln{5})=5$

    so on and so forth?

    Thanks.
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  2. #2
    MHF Contributor Amer's Avatar
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    u have a strange name
    anyway

    $\displaystyle log x = log_{10} x $ when u did not write the base it is 10

    $\displaystyle log_a a = 1 $ in general

    if u do not know what is the logarithm this is an example
    suppose $\displaystyle x^a = b $ then $\displaystyle a = \log _x b $

    so
    to find $\displaystyle 10^{\log 5} $

    let $\displaystyle x = 10^{\log_{10} 5} $

    $\displaystyle \log _{10} x = \log _{10} 5 $ so $\displaystyle x = 5 $

    second one same as the first one

    $\displaystyle ln = \log _{e} $
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  3. #3
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    lg means base 10 already. -.-
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  4. #4
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    Quote Originally Posted by Amer View Post
    u have a strange name
    anyway

    $\displaystyle log x = log_{10} x $ when u did not write the base it is 10

    $\displaystyle log_a a = 1 $ in general

    if u do not know what is the logarithm this is an example
    suppose $\displaystyle x^a = b $ then $\displaystyle a = \log _x b $

    so
    to find $\displaystyle 10^{\log 5} $

    let $\displaystyle x = 10^{\log_{10} 5} $

    $\displaystyle \log _{10} x = \log _{10} 5 $ so $\displaystyle x = 5 $

    second one same as the first one

    $\displaystyle ln = \log _{e} $
    Your explanation is unclear but u gave me inspiration to understand. THANKS
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  5. #5
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    Specifically, $\displaystyle log_a(x)$ or "logarithm to base a of x" is defined as the "inverse function" to $\displaystyle a^x$. That is, each "undoes" the other: $\displaystyle a^{log_a(x)}= x$ and $\displaystyle log_a(a^x)= x$.

    Your "lg" is an abbreviation for the "common logarithm" or "logarithm base 10", $\displaystyle log_{10}(x)$ which is the inverse to $\displaystyle 10^x$. $\displaystyle lg(10^x)= log_{10}(10^x)= x$ for all x (and so for x= 5 $\displaystyle lg(10^5)= 5$). $\displaystyle 10^{lg(x)}= 10^{log_{10}(x)}= x$ for all positive x (and so for x= 5 $\displaystyle 5^{lg(5)}= 5$.

    "ln" is an abbreviation for the "natural logarithm" or "logarithm base e" (e is about 2.718...) which is inverse to $\displaystyle e^x$. $\displaystyle ln(e^x)= log_e(e^x)= x$ for all x (and so for x= 5 $\displaystyle ln(e^5)= 5$. $\displaystyle e^{ln(x)}= x$ for all positive x (and so for x= 5 $\displaystyle e^{ln(x)}= x$.)

    (Note the difference between "for all x" and "for positive x". The function "$\displaystyle a^x$ is only defined for positive a and always has a positive value- that is, $\displaystyle a^x$ is a function from the set of all real numbers to the set of all positive real numbers. It's inverse, $\displaystyle ln_a(x)$, is a function from the set of all positive real numbers to the set of real numbers.)
    Last edited by mr fantastic; Sep 7th 2010 at 12:34 PM. Reason: Fixed a latex tag.
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  6. #6
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    how do you guys learn to write the formula source code? I actually trial and error. Is there anyway to learn?
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  7. #7
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    Quote Originally Posted by stupidguy View Post
    how do you guys learn to write the formula source code? I actually trial and error. Is there anyway to learn?
    It is LaTeX.
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  8. #8
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    Quote Originally Posted by Plato View Post
    It is LaTeX.
    So am I supposed to memorize the list of symbols and characters in order to use? Wouldn't that be too tedious?
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  9. #9
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    Quote Originally Posted by stupidguy View Post
    So am I supposed to memorize the list of symbols and characters in order to use? Wouldn't that be too tedious?
    It may be tedious, but it is well worth it.
    Posters who use LaTeX usually get faster and better help.
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