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10-log-3-6-15-log-3-2-3-6-log-3-2-3-5-log-3-4-3-

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Question Number 209991 by efronzo1 last updated on 28/Jul/24
((10^(log _3 (6)) . 15^(log _3 ((2/3))) )/(6^(log _3 ((2/3))) . 5^(log _3 ((4/3))) )) =?
$$\:\:\:\:\:\frac{\mathrm{10}^{\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{6}\right)} .\:\mathrm{15}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)} }{\mathrm{6}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)} .\:\mathrm{5}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{4}}{\mathrm{3}}\right)} }\:=?\: \\ $$
Answered by Rasheed.Sindhi last updated on 28/Jul/24
((10^(log _3 (6)) . 15^(log _3 ((2/3))) )/(6^(log _3 ((2/3))) . 5^(log _3 ((4/3))) )) =((2^(log_3 6 ) .5^(log_3 6 ) .3^(log_3 ((2/3)) ) .5^(log_3 ((2/3)) ) )/(2^(log_3 ((2/3)) ) .3^(log_3 ((2/3)) ) .5^(log_3 ((4/3)) ) )) =((2^(log_3 6 ) .5^(log_3 6 ) .5^(log_3 ((2/3)) ) )/(2^(log_3 ((2/3)) ) .5^(log_3 ((4/3)) ) )) =2^(log_3 2+log_3 3−(log_3 2−log_3 3) ) .5^(log_3 2−log_3 3−(log_3 4−log_3 3) ) =2^(log_3 2+log_3 3−log_3 2+log_3 3 ) .5^(log_3 6+ log_3 2−log_3 3−2log_3 2+log_3 3 ) =2^(log_3 3+log_3 3 ) .5^(log_3 2+log_3 3+ log_3 2−2log_3 2 ) =2^(2log_3 3 ) .5^(log_3 3 ) =2^(2(1)) .5^1 =4.5=20
$$\:\:\:\:\:\frac{\mathrm{10}^{\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{6}\right)} .\:\mathrm{15}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)} }{\mathrm{6}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)} .\:\mathrm{5}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{4}}{\mathrm{3}}\right)} }\: \\ $$$$=\frac{\mathrm{2}^{\mathrm{log}_{\mathrm{3}} \mathrm{6}\:} .\mathrm{5}^{\mathrm{log}_{\mathrm{3}} \mathrm{6}\:} .\cancel{\mathrm{3}^{\mathrm{log}_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)\:} }.\mathrm{5}^{\mathrm{log}_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)\:} }{\mathrm{2}^{\mathrm{log}_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)\:} .\cancel{\mathrm{3}^{\mathrm{log}_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)\:} }.\mathrm{5}^{\mathrm{log}_{\mathrm{3}} \left(\frac{\mathrm{4}}{\mathrm{3}}\right)\:} } \\ $$$$=\frac{\mathrm{2}^{\mathrm{log}_{\mathrm{3}} \mathrm{6}\:} .\mathrm{5}^{\mathrm{log}_{\mathrm{3}} \mathrm{6}\:} .\mathrm{5}^{\mathrm{log}_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)\:} }{\mathrm{2}^{\mathrm{log}_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)\:} .\mathrm{5}^{\mathrm{log}_{\mathrm{3}} \left(\frac{\mathrm{4}}{\mathrm{3}}\right)\:} } \\ $$$$=\mathrm{2}^{\mathrm{log}_{\mathrm{3}} \mathrm{2}+\mathrm{log}_{\mathrm{3}} \mathrm{3}−\left(\mathrm{log}_{\mathrm{3}} \mathrm{2}−\mathrm{log}_{\mathrm{3}} \mathrm{3}\right)\:} .\mathrm{5}^{\mathrm{log}_{\mathrm{3}} \mathrm{2}−\mathrm{log}_{\mathrm{3}} \mathrm{3}−\left(\mathrm{log}_{\mathrm{3}} \mathrm{4}−\mathrm{log}_{\mathrm{3}} \mathrm{3}\right)\:\:} \\ $$$$=\mathrm{2}^{\cancel{\mathrm{log}_{\mathrm{3}} \mathrm{2}}+\mathrm{log}_{\mathrm{3}} \mathrm{3}−\cancel{\mathrm{log}_{\mathrm{3}} \mathrm{2}}+\mathrm{log}_{\mathrm{3}} \mathrm{3}\:} .\mathrm{5}^{\mathrm{log}_{\mathrm{3}} \mathrm{6}+\:\mathrm{log}_{\mathrm{3}} \mathrm{2}−\cancel{\mathrm{log}_{\mathrm{3}} \mathrm{3}}−\mathrm{2log}_{\mathrm{3}} \mathrm{2}+\cancel{\mathrm{log}_{\mathrm{3}} \mathrm{3}}\:\:} \\ $$$$=\mathrm{2}^{\mathrm{log}_{\mathrm{3}} \mathrm{3}+\mathrm{log}_{\mathrm{3}} \mathrm{3}\:} .\mathrm{5}^{\mathrm{log}_{\mathrm{3}} \mathrm{2}+\mathrm{log}_{\mathrm{3}} \mathrm{3}+\:\:\mathrm{log}_{\mathrm{3}} \mathrm{2}−\mathrm{2log}_{\mathrm{3}} \mathrm{2}\:\:} \\ $$$$=\mathrm{2}^{\mathrm{2log}_{\mathrm{3}} \mathrm{3}\:} .\mathrm{5}^{\mathrm{log}_{\mathrm{3}} \mathrm{3}\:\:} \\ $$$$=\mathrm{2}^{\mathrm{2}\left(\mathrm{1}\right)} .\mathrm{5}^{\mathrm{1}} =\mathrm{4}.\mathrm{5}=\mathrm{20} \\ $$
Answered by alusto22 last updated on 28/Jul/24
= 6^(log _3 (10)−log _3 ((2/3))) . ((15^(log _3 ((2/3))) )/5^(log _3 ((4/3))) ) = 6^(log _3 (15)) .((15^(log _3 ((2/3))) )/5^(log _3 ((4/3))) ) = 15^(log _3 (6)+log _3 ((2/3))) .(1/5^(log _3 ((4/3))) ) = 15^(log _3 (4)) . 5^(−log _3 ((4/3))) = 5^(log _3 (3)) . 3^(log _3 (4)) = 20
$$\:\:=\:\mathrm{6}^{\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{10}\right)−\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)} .\:\frac{\mathrm{15}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)} }{\mathrm{5}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{4}}{\mathrm{3}}\right)} } \\ $$$$\:\:=\:\mathrm{6}^{\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{15}\right)} \:.\frac{\mathrm{15}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)} }{\mathrm{5}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{4}}{\mathrm{3}}\right)} } \\ $$$$\:\:\:=\:\mathrm{15}^{\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{6}\right)+\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)} \:.\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{4}}{\mathrm{3}}\right)} }\: \\ $$$$\:\:\:=\:\mathrm{15}^{\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{4}\right)} .\:\mathrm{5}^{−\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{4}}{\mathrm{3}}\right)} \\ $$$$\:\:\:=\:\mathrm{5}^{\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{3}\right)} \:.\:\mathrm{3}^{\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{4}\right)} \\ $$$$\:\:\:=\:\mathrm{20} \\ $$

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