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Math Help - [SOLVED] Solve Logaritmic Equation in terms of natural log

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    [SOLVED] Solve Logaritmic Equation in terms of natural log

    I need to solve the following equation:

    10^(2X+3) = 280

    To express in logarithmic, I know I need to do the following:

    log(10)^280 = 2X+3

    At this point I'm not sure where to go. I can move the three over to the left:

    (log(10)^280) - 3 = 2X

    Then divide by two:

    ((log(10)^280) - 3))/2 = X

    I get -.276 = X, but I feel like I'm missing a step in determining the answer. Did I do the steps correctly to come to this answer or is something missing?
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    Quote Originally Posted by Snowcrash View Post
    I need to solve the following equation:

    10^(2X+3) = 280

    To express in logarithmic, I know I need to do the following:

    log(10)^280 = 2X+3

    At this point I'm not sure where to go. I can move the three over to the left:

    (log(10)^280) - 3 = 2X

    Then divide by two:

    ((log(10)^280) - 3))/2 = X

    I get -.276 = X, but I feel like I'm missing a step in determining the answer. Did I do the steps correctly to come to this answer or is something missing?
    10^{2x + 3} = 280

    \ln{10^{2x + 3}} = \ln{280}

    (2x + 3)\ln{10} = \ln{280}

    2x + 3 = \frac{\ln{280}}{\ln{10}}

    2x = \frac{\ln{280}}{\ln{10}} - 3

    x = \frac{1}{2}\left(\frac{\ln{280}}{\ln{10}} - 3\right).
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    Great - I see I did not use the natural logarithmic form, but the answer is the same. (.276) Thank you!
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    Quote Originally Posted by Snowcrash View Post
    Great - I see I did not use the natural logarithmic form, but the answer is the same. (.276) Thank you!
    No problem.

    My personal preference is always the natural logarithm, but in this case, your answer might look a bit neater if you use the logarithm with base 10.


    10^{2x + 3} = 280

    10^{2x + 3} = 28\cdot 10

    \log_{10}{10^{2x + 3}} = \log_{10}{(28\cdot 10)}

    (2x + 3)\log_{10}{10} = \log_{10}{28} + \log_{10}{10}

    2x + 3 = \log_{10}{28} + 1

    2x = \log_{10}{28} - 2

    x = \frac{1}{2}\log_{10}{28} - 1.
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