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2-a-3-b-36-c-then-prove-that-ab-2c-a-b-




Question Number 191839 by MATHEMATICSAM last updated on 01/May/23
2^a  = 3^b  = 36^c  then prove that  ab = 2c(a + b).
$$\mathrm{2}^{{a}} \:=\:\mathrm{3}^{{b}} \:=\:\mathrm{36}^{{c}} \:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$${ab}\:=\:\mathrm{2}{c}\left({a}\:+\:{b}\right). \\ $$
Answered by AST last updated on 01/May/23
a=blog_2 (3)⇒(a/b)=log_2 (3)⇒((a+b)/b)=log_2 (6)  a=clog_2 (36)⇒(a/c)=log_2 (36)⇒(a/(2c))=log_2 (6)  ⇒(a/(2c))=((a+b)/b)⇒ab=2c(a+b)                                          □
$${a}={blog}_{\mathrm{2}} \left(\mathrm{3}\right)\Rightarrow\frac{{a}}{{b}}={log}_{\mathrm{2}} \left(\mathrm{3}\right)\Rightarrow\frac{{a}+{b}}{{b}}={log}_{\mathrm{2}} \left(\mathrm{6}\right) \\ $$$${a}={clog}_{\mathrm{2}} \left(\mathrm{36}\right)\Rightarrow\frac{{a}}{{c}}={log}_{\mathrm{2}} \left(\mathrm{36}\right)\Rightarrow\frac{{a}}{\mathrm{2}{c}}={log}_{\mathrm{2}} \left(\mathrm{6}\right) \\ $$$$\Rightarrow\frac{{a}}{\mathrm{2}{c}}=\frac{{a}+{b}}{{b}}\Rightarrow{ab}=\mathrm{2}{c}\left({a}+{b}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Box \\ $$

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