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Question Number 113426 by bemath last updated on 13/Sep/20

	Answered by bemath last updated on 13/Sep/20
	$(1) 3^(3b) = 5^(4a) →log _3 (3^(3b) )=log _3 (5^(4a) ) (2) 3^(2b+2) = 5^(3a) →log _3 (3^(2b+2) )=log _3 (5^(3a) ) { ((3b=log _3 (5^(4a) ))),((2b+2=log _3 (5^(3a) ))) :} →2(((log _3 (5^(4a) ))/3))+2=log _3 (5^(3a) ) →2(log _3 (5^(4a) ))=3log _3 (5^(3a) )−6 →log _3 (5^(8a) )=log _3 ((5^(9a) /3^6 )) →3^6 = 5^a . Now →9^(b+1) = 125^a = (5^a )^3 → 3^(2b+2) = (3^6 )^3 =3^(18) → 2b+2 = 18 ; b = 8 and 9^9 = (5^(3a) )→3aln (5)=18ln (3) ⇒a = ((6 ln (3))/(ln (5))) = 6.log _5 (3)$
	$(1) 3^{3 b} = 5^{4 a} \to \log_{3} (3^{3 b}) = \log_{3} (5^{4 a})$ $(2) 3^{2 b + 2} = 5^{3 a} \to \log_{3} (3^{2 b + 2}) = \log_{3} (5^{3 a})$ ${\begin{cases} 3 b = \log_{3} (5^{4 a}) \\ 2 b + 2 = \log_{3} (5^{3 a}) \end{cases}$ $\to 2 (\frac{\log_{3} (5^{4 a})}{3}) + 2 = \log_{3} (5^{3 a})$ $\to 2 (\log_{3} (5^{4 a})) = 3 \log_{3} (5^{3 a}) - 6$ $\to \log_{3} (5^{8 a}) = \log_{3} (\frac{5^{9 a}}{3^{6}})$ $\to 3^{6} = 5^{a} . N o w$ $\to 9^{b + 1} = 125^{a} = {(5^{a})}^{3}$ $\to 3^{2 b + 2} = {(3^{6})}^{3} = 3^{18}$ $\to 2 b + 2 = 18; b = 8$ $a n d 9^{9} = (5^{3 a}) \to 3 a \ln (5) = 18 \ln (3)$ $\Rightarrow a = \frac{6 \ln (3)}{\ln (5)} = 6 . \log_{5} (3)$
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