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Thread: Logarithms and surds

  1. #1
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    Logarithms and surds

    Thanks for all your help. I appreciate it a lot.

    Question 1
    If $\displaystyle a^2 + b^2 = 7ab$ show that

    $\displaystyle 2 \log_{10} \frac{a+b}{3} = \log_{10} a + \log_{10} b $

    Question 2
    If $\displaystyle 1 + \log_{a} (7x - 3a) = 2 \log_{a} x + \log_{a} 2 $ find in terms of a the possible values of x
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  2. #2
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    Quote Originally Posted by ecogreen View Post
    Thanks for all your help. I appreciate it a lot.

    Question 1
    If $\displaystyle a^2 + b^2 = 7ab$ show that

    $\displaystyle 2 \log_{10} \frac{a+b}{3} = \log_{10} a + \log_{10} b $

    Question 2
    If $\displaystyle 1 + \log_{a} (7x - 3a) = 2 \log_{a} x + \log_{a} 2 $ find in terms of a the possible values of x
    Q.1 $\displaystyle a^2 + b^2 = 7ab$

    Add 2ab to both the sides. Then

    $\displaystyle a^2 + b^2 + 2ab = 9ab$

    $\displaystyle (a+b)^2 = 9ab$

    $\displaystyle (\frac{a+b}{3})^2 = ab$

    Now take log on both the side to the base 10.

    Q.2

    $\displaystyle 1 + \log_{a} (7x - 3a) = 2 \log_{a} x + \log_{a} 2 $


    $\displaystyle \log_{a}(a) + \log_{a} (7x - 3a) = \log_{a}2x^2$


    $\displaystyle \log_{a}a(7x - 3a) = \log_{a}2x^2$

    $\displaystyle 2x^2 - 7ax + 3a^2 = 0$

    Now solve the quadratic to find x in terms of a.
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