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3-2-log-9-25-5-9-2-x-

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Question Number 4515 by love math last updated on 04/Feb/16
3^(2+log_9 25) =5×9^(2/x)
32+log925=5×92x
Commented by love math last updated on 05/Feb/16
Is there any way to prove 3^(2+log_9 25) =45 without using calculator?
Isthereanywaytoprove32+log925=45withoutusingcalculator?
Commented by Rasheed Soomro last updated on 06/Feb/16
3^(2+log_9 25) =45 LHS: 3^(2+log_9 25) =3^2 .3^(log_9 25) =9×3^((log 25)/(log 9)) =9×3^((log 5^2 )/(log 3^2 )) =9×3^((2log 5)/(2log 3)) =9×3^((log 5)/(log 3)) =9×3^(log_3 5) =9×5=45=RHS ......... Let y=3^(log_3 5) log_3 y=log_3 (3^(log_3 5) ) =(log_3 5) log_3 3 =(log_3 5)(1) =log_3 5 log_3 y=log_3 5 y=5 ∴ 3^(log_3 5) =5
32+log925=45LHS:32+log925=32.3log925=9×3log25log9=9×3log52log32=9×32log52log3=9×3log5log3=9×3log35=9×5=45=RHS………Lety=3log35log3y=log3(3log35)=(log35)log33=(log35)(1)=log35log3y=log35y=5∴3log35=5
Answered by Rasheed Soomro last updated on 05/Feb/16
3^(2+log_9 25) =5×9^(2/x) In order to write 5 as power of 3 : let 3^y =5 ylog 3=log 5 y=((log 5)/(log 3)) 3^(2+log_9 25) =3^((log 5)/(log 3)) ×3^(4/x) 3^(2+log_9 25) =3^(((log 5)/(log 3))+(4/x)) ((log 5)/(log 3))+(4/x)=2+log_9 25 ((log 5)/(log 3))+(4/x)=2+((log 25)/(log 9)) ((log 5)/(log 3))+(4/x)=2+((log 5^2 )/(log 3^2 )) ((log 5)/(log 3))+(4/x)=2+((2log 5)/(2log 3)) ((log 5)/(log 3))+(4/x)=2+((log 5)/(log 3)) (4/x)=2 x=2
32+log925=5×92xInordertowrite5aspowerof3:let3y=5ylog3=log5y=log5log332+log925=3log5log3×34x32+log925=3log5log3+4xlog5log3+4x=2+log925log5log3+4x=2+log25log9log5log3+4x=2+log52log32log5log3+4x=2+2log52log3log5log3+4x=2+log5log34x=2x=2

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