1-2-log-4-36-log-6-64-

Question Number 177360 by jlewis last updated on 04/Oct/22

1 / 2 {l o g}_{4} 36 \times {l o g}_{6} 64

Answered by TheHoneyCat last updated on 08/Oct/22

= \frac{1}{2} \log_{4} (4 \times 9) \times \log_{6} (6 \times 9)

= \frac{1}{2} (\log_{4} 4 + \log_{4} 9) (\log_{6} 6 + \log_{6} 9)

= \frac{1}{2} (1 + \frac{\ln 9}{\ln 4}) (1 + \frac{\ln 9}{\ln 6})

= \frac{1}{2} (1 + \frac{2 \ln 3}{2 \ln 2}) (1 + \frac{2 \ln 3}{\ln 2 + \ln 3})

= \frac{1}{2} (1 + \frac{\ln 3}{\ln 2}) (1 + \frac{2 \ln 3}{\ln 2 + \ln 3})

= \frac{1}{2} \times \frac{\ln 2 + \ln 3}{\ln 2} \times \frac{\ln 2 + 3 \ln 3}{\ln 2 + \ln 3}

= \frac{\ln 2 + 3 \ln 3}{2 \ln 2}_{}