Question

cum se rezolva X la lg(2x) = 5

Answer 1

$x^{lg(2x)}=5$

Vom  logaritma si vom obtine:

$lg(x^{lg(2x)})=lg5$

Puterea logaritmului coboara in fata si obtinem:

lg(2x)×lg(x)=lg5

lg(2x)=lg2+lg(x)

Vom avea:

(lg2+lg(x))×lg(x)=lg5

Notam lg(x)=y

lg2×y+y²=lg5

Stim ca lg5=lg(10:2)=lg10-lg2=1-lg2

lg2×y+y²=lg10-lg2

lg2×y+y²=1-lg2

Grupam convenabil termenii si obtinem:

lg2(y+1)+y²-1=0

lg2(y+1)+(y+1)(y-1)=0

Dam factor comun y+1 si obtinem:

(y+1)(lg2+y-1)=0

Avem doua solutii:

1) y+1=0

y=-1

lgx=-1

x=10⁻¹

2) lg2+y-1=0

y=1-lg2

y=lg5

lgx=lg5

x=5