设3^a=4^b=36,求2/a+1/b的值

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设3^a=4^b=36,求2/a+1/b的值

设3^a=4^b=36,求2/a+1/b的值
设3^a=4^b=36,求2/a+1/b的值

设3^a=4^b=36,求2/a+1/b的值
3^a=4^b=36
==>a=log3 36=log3 4+log3 9=2log3 2+2
==>b=log4 36=log4 4+log4 9=1+log2 3
==>2/a+1/b=1/(1+log3 2)+1/(1+log2 3)=lg3/(lg2+lg3)+lg2/(lg2+lg3)=1

因为a3^a=36,4^b=36。所以:
a=lg36/lg3=2lg6/lg3,
b=lg36/lg4=lg6/lg2
2/a+1/b
=(a+2b)/(ab)
=(2lg6/lg3+2lg6/lg2)/[2lg6×lg6/(gl2×lg3)]
=[2lg6(lg2+lg3)/lg2×lg3]×[lg2×lg3/(2lg6×lg6)]
=2lg6(lg2+lg3)/(2lg6×lg6)
=(lg2+lg3)/lg6
=lg6/lg6
=1