Question

1．計(jì)算下列各式的值．
（1）$\frac{lg2+lg5-lg8}{lg50-lg40}$；
（2）log₅35-2log₅$\frac{7}{3}$+log₅7-log₅1.8；
（3）$\frac{lg\sqrt{27}+lg8-lg\sqrt{1000}}{lg1.2}$；
（4）2（lg$\sqrt{2}$）²+lg$\sqrt{2}$•lg5+$\sqrt{（lg\sqrt{2}）^{2}-lg2+1}$．

Answer 1

分析 利用對(duì)數(shù)的性質(zhì)和運(yùn)算法則求解．

解答 解：（1）$\frac{lg2+lg5-lg8}{lg50-lg40}$
=$\frac{lg\frac{2×5}{8}}{lg\frac{50}{40}}$
=$\frac{lg\frac{5}{4}}{lg\frac{5}{4}}$=1．
（2）log₅35-2log₅$\frac{7}{3}$+log₅7-log₅1.8
=$lo{g}_{5}（35×\frac{9}{49}×7×\frac{5}{9}）$
=log₅25
=2．
（3）$\frac{lg\sqrt{27}+lg8-lg\sqrt{1000}}{lg1.2}$
=$\frac{\frac{3}{2}lg3+3lg2-\frac{3}{2}}{lg1.2}$
=$\frac{\frac{3}{2}（lg3+lg4-lg10）}{lg1.2}$
=$\frac{\frac{3}{2}lg1.2}{lg1.2}$
=$\frac{3}{2}$．
（4）2（lg$\sqrt{2}$）²+lg$\sqrt{2}$•lg5+$\sqrt{（lg\sqrt{2}）^{2}-lg2+1}$
=lg$\sqrt{2}$（2lg$\sqrt{2}$+lg5）+1-lg$\sqrt{2}$
=lg$\sqrt{2}$（lg2+lg5）+1-lg$\sqrt{2}$
=lg$\sqrt{2}$+1-lg$\sqrt{2}$
=1．

點(diǎn)評(píng) 本題考查對(duì)數(shù)式的化簡(jiǎn)求值，是基礎(chǔ)題，解題時(shí)要認(rèn)真審題，注意對(duì)數(shù)的性質(zhì)和運(yùn)算法則的合理運(yùn)用．