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Math Help - Simplifying an expression with e

  1. #1
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    Simplifying an expression with e

    Simplify the expression e^{(\ln3) +2 \ln x}.

    My work (I feel like it's not legal!):

    \ \ln(e^{ln3+2 \ln x})
    = \ln3+2 \ln x
    = 2\ln (3x)
    = \ln (3x^2)
    = e^{ln3x^2}
    = 3x^2

    Am I allowed to just take the natural log of an expression when simplifying since there's no other side to take the natural log of? Same question for bumping the expression up into the exponent of e?
    Last edited by mr fantastic; March 20th 2011 at 06:54 PM. Reason: Fixed and clarified latex.
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  2. #2
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    Quote Originally Posted by lfroehli View Post
    Simplify the expression e^{\ln3+2\lnx}.

    My work (I feel like it's not legal!):

    \ \ln(e^{ln3+2\lnx})
    = \ln3+2\lnx
    = 2\ln3x
    = \ln3x^2
    = e^{ln3x^2}
    = 3x^2

    Am I allowed to just take the natural log of an expression when simplifying since there's no other side to take the natural log of? Same question for bumping the expression up into the exponent of e?
    No, you can't.

    \displaystyle e^{ln(3) + 2} = e^{ln(3)} \cdot e^2

    Does this help?

    -Dan

    Wait. where did the x come from?
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  3. #3
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    ah, I messed up the question. It's simplify the expression e^{\ln3 + 2 \ln{x}}.
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  4. #4
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    Quote Originally Posted by topsquark View Post
    No, you can't.

    \displaystyle e^{ln(3) + 2} = e^{ln(3)} \cdot e^2

    Does this help?

    -Dan

    Wait. where did the x come from?
    The OP did not use the latex commands correctly and so the x did not show up. I have fixed this.
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  5. #5
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    First simplify the exponent into a single logarithm...

    \displaystyle \ln{(3)} + 2\ln{(x)} = \ln{(3)} + \ln{\left(x^2\right)} = \ln{(3x^2)}.


    Then \displaystyle e^{\ln{(3)} + 2\ln{(x)}} = e^{\ln{\left(3x^2\right)}}. The rest of the solution should be obvious...
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  6. #6
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    Quote Originally Posted by lfroehli View Post
    Simplify the expression e^{(\ln3) +2 \ln x}.

    My work (I feel like it's not legal!):

    \ \ln(e^{ln3+2 \ln x})
    = \ln3+2 \ln x
    This is alright if you remember that you took the log and reverse it at the end.

    = 2\ln (3x)
    But this is NOT right. The 2 is multiplying only ln(x), not log 3. What you need to do is take that 2 inside the ln:
    ln 3+ 2ln x= ln 3+ ln x^2= ln (3x^2)

    = \ln (3x^2)
    Okay, now you have fixed the "2" problem!

    = e^{ln3x^2}
    = 3x^2
    Good! Now you have reversed the ln you did at the beginning and have the correct answer.

    Am I allowed to just take the natural log of an expression when simplifying since there's no other side to take the natural log of? Same question for bumping the expression up into the exponent of e?
    What you did is correct but it would have been simpler to use the logarithm properties in the exponent.
    e^{ln(3)+ 2ln(x)}= e^{ln(3)+ ln(x^2)}= e^{ln(3x^2)}= 3x^2.
    Last edited by HallsofIvy; March 21st 2011 at 11:10 AM.
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