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If-log-12-18-A-and-log-24-54-B-then-prove-that-AB-5-A-B-1-

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Question Number 202172 by MATHEMATICSAM last updated on 22/Dec/23
If log_(12) 18 = A and log_(24) 54 = B then prove that AB + 5(A − B) = 1.
Iflog1218=Aandlog2454=BthenprovethatAB+5(A−B)=1.
Answered by Rasheed.Sindhi last updated on 22/Dec/23
12^A =18 & 24^B =54 (2^2 .3)^A =18 & (2^3 .3)^B =54 2^(2A) .3^A =2.3^2 & 2^(3B) .3^B =2.3^3 ((2^(2A) .3^A )/(2^(3B) .3^B ))=((2.3^2 )/(2.3^3 ))=3^(−1) 2^(2A−3B) .3^(A−B) =2^0 .3^(−1) 2A−3B=0 ∧ A−B=−1(⇒A=B−1) 2(B−1)−3B=0 B=−2⇒A=−2−1=−3 AB + 5(A − B) = 1 ⇒AB + 5A − 5B−25+25=1 A(B+5)−5(B+5)+25=1 (A−5)(B+5)+25=1 (−3−5)(−2+5)+25=1 (−8)(3)+25=1 1=1 QED
12A=18&24B=54(22.3)A=18&(23.3)B=5422A.3A=2.32&23B.3B=2.3322A.3A23B.3B=2.322.33=3−122A−3B.3A−B=20.3−12A−3B=0∧A−B=−1(⇒A=B−1)2(B−1)−3B=0B=−2⇒A=−2−1=−3AB+5(A−B)=1⇒AB+5A−5B−25+25=1A(B+5)−5(B+5)+25=1(A−5)(B+5)+25=1(−3−5)(−2+5)+25=1(−8)(3)+25=11=1QED

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