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Math Help - Logarithmic Inequalities.

  1. #1
    Junior Member Kaloda's Avatar
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    Logarithmic Inequalities.

    For which real numbers x does inequality 2\log_x\frac{a+b}{2}\leq\log_x(a)+\log_x(b)
    holds for all positive numbers a and b?
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  2. #2
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    Re: Logarithmic Inequalities.

    Quote Originally Posted by Kaloda View Post
    For which real numbers x does inequality 2\log_x\frac{a+b}{2}\leq\log_x(a)+\log_x(b)
    holds for all positive numbers a and b?

    Here are some observations. For \log_x(t) to be defined it is necessary for t>0 and 0<x<1\text{ or }x>1.

    The function \log_x(t) is one-to-one and it is increasing if x>1 and decreasing for 0<x<1.

    So this is a concept question.
    Compare \left(\frac{a+b}{2}\right)^2\le\text{ or }\ge a\cdot b.
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