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Math Help - Comparing Two Logarithms Base 10

  1. #1
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    Comparing Two Logarithms Base 10

    Hi all ---

    For this multiple choice question, I don't know how to compare (a) and (b).



    I try --- \log \pi ? \sqrt{\log (\pi)^2}

    Then --- \log \pi ? \sqrt{2\log\pi}

    or if I sub in L --- L ? \sqrt{2L}

    Squaring both sides --- L^2 ? 2L

    Because 0 < L < 1 --- squaring this for the 1st and then multiplying by 2 for the 2nd --- 0 < L^2 < 1 and 0 < 2L < 2.

    1st Question --- But how can I compare the last two inequalities? I don't know what specifically L is?

    2nd Question --- How does solution get \sqrt{2L} > \sqrt{L \times L}?

    Thanks a lot ---
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  2. #2
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    Re: Comparing Two Logarithms Base 10

    L = \log_{10}{\pi} < 1

    simplified choices in terms of L ...

    (a) L (b) \sqrt{2L} (c) \frac{1}{L^3} (d) \frac{2}{L}

    for (a) and (b) ...

    L = \sqrt{L \cdot L} < \sqrt{2 \cdot L} because 2 > L

    finally, since L < 1 , both (c) and (d) are > 1 > L ... you don't need to compare them against each other.
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