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

Question Number 147334 by Tawa11 last updated on 19/Jul/21

If \log_{6} 30 = a, \log_{15} 24 = b, evaluate : \log_{12} 60

Commented by otchereabdullai@gmail.com last updated on 20/Jul/21

nice!

Answered by bramlexs22 last updated on 20/Jul/21

If {\begin{cases} \log_{6} (30) = a \\ \log_{15} (24) = b \end{cases} Evaluate \log_{12} (60) .

\equiv solution \equiv

(∙) a = \log_{6} (6 \times 5) = 1 + \log_{6} (5)

\Rightarrow \log_{6} (5) = a - 1

(∙) b = \log_{15} (24) = \frac{\log_{6} (24)}{\log_{6} (15)} = \frac{1 + {2 \log}_{6} (2)}{a - 1 + \log_{6} (3)}

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

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

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

(∙) \log_{12} (60) = \frac{1 + \log_{6} (5) + \log_{6} (2)}{1 + \log_{6} (2)}

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

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

Commented by Tawa11 last updated on 20/Jul/21

Thanks sir . God bless you .

Facebook Tweet Pin