Question

求值:?

(1);?

(2)lg5·lg20+lg²2;?

(3)已知log₂3=a,3 ^b=7,求log₁₂56的值.?

Answer 1

解：(1)(2)嚴格按照指數(shù)、對數(shù)的運算法則計算,(3)先將3 ^b=7轉(zhuǎn)化為log₃7=b,然后設(shè)法將log₁₂56化成關(guān)于log₂3和log₃7的表達式即可求值.?

(1) = =.

(2)lg5·lg20+lg²2=lg5(lg4+lg5)+lg²2=2lg2·lg5+lg²5+lg²2=(lg2+lg5) ²=1.?

(3)解法一:

∵log₂3=a,∴2 ^a=3.

又3 ^b=7,∴7=(2 ^a) ^b=2 ^ab.

故56=2 ^3+ab.

又12=3·4=2 ^a·4=2 ^a+2,

從而.故.

解法二:

∵,∴.

又3 ^b=7,∴l(xiāng)og ₃7=b.從而====

=.

解法三:

∵log₂3==a,∴l(xiāng)g3=alg2.

又3 ^b=7,∴l(xiāng)g7=blg3.

∴l(xiāng)g7=ablg2.從而log₁₂56= == =.