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

Question Number 199389 by depressiveshrek last updated on 02/Nov/23

\log_{12} 60 = ?

\log_{6} 30 = a

\log_{15} 24 = b

Answered by cortano12 last updated on 03/Nov/23

\log_{12} (60) = \frac{\log_{6} (6.5.2)}{\log_{6} (6.2)}

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

Commented by depressiveshrek last updated on 03/Nov/23

I {don}^{'} t understand how you got there \dots

Answered by mr W last updated on 04/Nov/23

\log_{15} 24 = \frac{\log_{6} 24}{\log_{6} 15} = \frac{\log_{6} 6 + {2 \log}_{6} 2}{\log_{6} 30 - \log_{6} 2}

= \frac{1 + {2 \log}_{6} 2}{\log_{6} 30 - \log_{6} 2}

= \frac{1 + {2 \log}_{6} 2}{a - \log_{6} 2} = b

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

\log_{12} 60

= \frac{\log_{6} 30 + \log_{6} 2}{\log_{6} 6 + \log_{6} 2}

= \frac{\log_{6} 30 + \log_{6} 2}{1 + \log_{6} 2}

= \frac{a + \frac{a b - 1}{2 + b}}{1 + \frac{a b - 1}{2 + b}}

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

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