Nilai dari ²log 3. log 8. 5log 6 log 5= ....​

Soal

Nilai dari ²log 3. log 8. 5log 6 log 5= ....​

Untuk menyelesaikan soal ini, kita dapat menggunakan beberapa sifat logaritma:

$$\begin{aligned}2\log 3.\log 8.5\log 6.\log 5 &= \log(3^2) + \log(8\log 8) + \log(6^5) + \log(5\log 5)\\ &= \log(9) + \log(8^{\log 8}) + \log(6^5) + \log(5^{\log 5})\\ &= \log(9\times8^{\log 8}\times6^5\times5^{\log 5})\\ &= \log(9\times2^{18}\times6^5\times5^{\log 5})\\ &= \log\left[\frac{9\times2^{18}\times6^5\times5^{\log 5}}{10^{\log(9\times2^{18}\times6^5\times5^{\log 5})}}\right]\\ &= \log\left[\frac{9\times2^{18}\times6^5\times5^{\log 5}}{(10^{\log 9})(10^{18})(10^{\log 6})(10^5)(10^{\log 5})}\right]\\ &= \log\left[\frac{9\times2^{18}\times6^5\times5^{\log 5}}{(9\times6\times5^2)}\right]\\ &= \log\left[\frac{2^{18}\times5^{\log 5-2}\times3^4}{2}\right]\\ &= \boxed{17\log 2 + (\log 5 - 2)\log 5 + 4\log 3}.\\ \end{aligned}$$