# log3 512 log43log38 log49

${\mathrm{log}}_{3}512.{\mathrm{log}}_{4}3\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{log}}_{2}512}{{\mathrm{log}}_{2}3}×\frac{{\mathrm{log}}_{2}3}{{\mathrm{log}}_{2}4}$

$=\frac{{\mathrm{log}}_{2}512}{{\mathrm{log}}_{2}4}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{9{\mathrm{log}}_{2}2}{2{\mathrm{log}}_{2}2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{9}{2}$

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${\mathrm{log}}_{3}8.{\mathrm{log}}_{4}9\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{log}}_{2}8}{{\mathrm{log}}_{2}3}×\frac{{\mathrm{log}}_{2}9}{{\mathrm{log}}_{2}4}$

$=\frac{{\mathrm{log}}_{2}\left({2}^{3}\right)}{{\mathrm{log}}_{2}3}×\frac{{\mathrm{log}}_{2}\left({3}^{2}\right)}{{\mathrm{log}}_{2}\left({2}^{2}\right)}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{3{\mathrm{log}}_{2}2}{{\mathrm{log}}_{2}3}×\frac{2{\mathrm{log}}_{2}3}{2{\mathrm{log}}_{2}2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=3$

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