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Math Help - Solving equation with negative exponential with logarithm

  1. #1
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    Solving equation with negative exponential with logarithm

    Hi guys!

    I'm struggling with this problem, I'm sure that the solution is not too difficult, but I just can't figure it out.

    6-2x 15 = 0. Now it is pretty easy to solve that equation the "regular" way and come to the answer of 540, but how do I solve it by using logarithm? If the exponential was positive that would also be easy, but it being negative is really screwing with me. Thanks for the help.
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  2. #2
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    Re: Solving equation with negative exponential with logarithm

    I'm pretty sure that \displaystyle \begin{align*} 6^{-2 \cdot 540} - 15 \end{align*} does NOT equal 0...

    \displaystyle \begin{align*} 6^{-2x} - 15 &= 0 \\ 6^{-2x} &= 15 \\ -2x &= \log_6{(15)} \\ x &= -\frac{1}{2}\log_6{(15)} \end{align*}
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    Re: Solving equation with negative exponential with logarithm

    Hi ManIsuck!

    6^{-2x} - 15 = 0

    6^{-2x} = 15

    \ln (6^{-2x}) = \ln 15

    (-2x) \ln 6 = \ln 15

    Can you take it from there?
    As you can see, the logarithm is never taken from a negative number.

    Btw, do you know which rules apply to calculating with logarithms?
    Because if you don't that makes it that much harder.
    In this particular calculation I used that:
    \ln(x^y) = y \ln x
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  4. #4
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    Re: Solving equation with negative exponential with logarithm

    I must be missing something. If the "answer" is the value of x that makes the equation true, then I can see at a glance that it must be a number between -1 and 0. How do you get 540? Can you tell me what is the "regular" way? Using natural logarithms, I find x is approximately -0.75570. See the attachment.
    Attached Files Attached Files
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  5. #5
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    Re: Solving equation with negative exponential with logarithm

    Thanks for the help guys. I'm not sure where the difference is coming from. I ran the equation through wolfram to check the answer. Maybe I'm using some sign in a different way than you guys?
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  6. #6
    Super Member ILikeSerena's Avatar
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    Re: Solving equation with negative exponential with logarithm

    Can you add a couple of parentheses around (-2x)?
    Afterward, check how your "input" looks and whether it matches your problem statement.
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  7. #7
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    Re: Solving equation with negative exponential with logarithm

    Thanks. Certainly seems like I picked the right nick. :/
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