Question

در صورتی که $$a,b,c,d$$ عددی طبیعی باشند و: $$log_ab= \frac{3}{2} \wedge log_cd= \frac{5}{4} \wedge a-c=9 \Longrightarrow b-d=?$$

Answer 1

$$ \frac{logb}{loga} = \frac{3}{2} \wedge \frac{logd}{logc} = \frac{5}{4} \Longrightarrow \frac{logb}{3} = \frac{loga}{2} =k \wedge \frac{logd}{5} = \frac{logc}{4} =p \Longrightarrow a= 10^{2k} \wedge b= 10^{3k} \wedge c= 10^{4p} \wedge d= 10^{5p} \wedge a-c=9 \Longrightarrow 10^{2k} - 10^{4p} =9 \Longrightarrow ( 10^{k} + 10^{2p} )( 10^{k} - 10^{2p} )=9 \Longrightarrow 10^{k} + 10^{2p} =9 \wedge 10^{k} - 10^{2p} =1 \Longrightarrow 10^{k} =5 \wedge (10^{2p} =4 \Longrightarrow 10^{p} =2) \Longrightarrow b= 5^{3} =125 \wedge d= 2^{5}=32 \Longrightarrow b-d=93 $$