Question

Valoarea expresiei E = lg^240 + lg^2 2/5 - 2lg 20*lg 0,0016 este egala cu:
Exercițiul 4

Answer 1

$\it lg40=lg4\cdot10=lg4+lg10=lg4+1 \\ \\ lg\dfrac{2}{5}=lg\dfrac{4}{10}=lg4\cdot10^{-1}=lg4+lg10^{-1}=lg4-1\\ \\ 2lg20=lg20^2=lg400=lg4\cdot100=lg4+lg100=lg4+2\\ \\ lg0,0016=lg\dfrac{16}{10000}=lg\dfrac{4^2}{10^4}=lg4^2\cdot10^{-4}=lg4^2+lg10^{-4}=2lg4-4=\\ \\ =2(lg4-2)=(lg4-2)\cdot2$

Notăm lg4 = a și expresia devine:

$\it E = (a+1)^2+(a-1)^2-(a+2)(a-2)\cdot2=a^2+2a+1+a^2-2a+1-(a^2-4)\cdot2=\\ \\ =2a^2+2-2a^2+8=10$