if-log-6-30-a-and-log-24-15-b-log-12-60-

Question Number 92010 by Rio Michael last updated on 04/May/20

if \log_{6} 30 = a and \log_{24} 15 = b

\log_{12} 60 = ?

Answered by jagoll last updated on 04/May/20

\log_{12} (60) = \frac{\log_{6} (30) + \log_{6} (2)}{1 + \log_{6} (2)}

= \frac{a + \log_{6} (2)}{1 + \log_{6} (2)} = \frac{a + (\frac{a - b}{2 b + 1})}{1 + (\frac{a - b}{2 b + 1})}

= \frac{a (2 b + 1) + a - b}{2 b + 1 + a - b} = \frac{2 a + 2 ab - b}{a + b + 1}

(1) \log_{24} (15) = \frac{\log_{6} (30) - \log_{6} (2)}{1 + 2 \log_{6} (2)} = b

a - \log_{6} (2) = b + 2 b \log_{6} (2)

(2 b + 1) \log_{6} (2) = a - b

\log_{6} (2) = \frac{a - b}{2 b + 1}

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