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Math Help - 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

    10^(lg{5})=5
    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

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

    log_a a = 1 in general

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

    so
    to find 10^{\log 5}

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

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

    second one same as the first one

    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

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

    log_a a = 1 in general

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

    so
    to find 10^{\log 5}

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

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

    second one same as the first one

    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, log_a(x) or "logarithm to base a of x" is defined as the "inverse function" to a^x. That is, each "undoes" the other: a^{log_a(x)}= x and log_a(a^x)= x.

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

    (Note the difference between "for all x" and "for positive x". The function " a^x is only defined for positive a and always has a positive value- that is, a^x is a function from the set of all real numbers to the set of all positive real numbers. It's inverse, 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; September 7th 2010 at 01:34 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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