Question

若log8a+log4b^2=5,且log8b+log4a^2=7,那么log2(ab)的值为

若log8a+log4b^2=5，且log8b+log4a^2=7，那么log2（ab）的值为

Answer 1

log8a+log4b^2=5
log2^3a+log2^2b^2=5
1/3log2a+2/2log2b=5
log2a/3+log2b=5
log2b=5-log2a/3 1

log8b+log4a^2=7
log2^3b+log2^2a^2=7
log2b/3+log2a=7 2
1代入2得
(5-log2a/3 )/3+log2a=7
5/3-log2a/9+log2a=7 两边乘9得
15-log2a+9log2a=63
8log2a=63-15=48
log2a=6 3
3代入1得
log2b=5-6/3=3
log2(ab)=log2a+log2b=6+3=9

Answer 2

log8a+log4b^2=5
log8b+log4a^2=7
变形，得
(1/3)log2(a)+log2(b)=5 (1)
(1/3)log2(b)+log2(a)=7 (2)
(1)+(2)
(1/3)[log2(a)+log2(b)]+log2(a)+log2(b)=12
(4/3)log2(ab)=12
log2(ab)=12/(4/3)=9