log(3x + 5) (6x2 +19x + 15) + log(2x + 3) (9x2 + 30x +25) = 4 - Maths - Logs Equations and Inequalities

Question

log(3x + 5) (6x2 +19x + 15) + log(2x +
3) (9x2 + 30x +25) = 4

Lovina Kansal · Accepted Answer

Dear student
log3x+56x2+19x+15+log2x+39x2+30x+25=4⇒log3x+52x+33x+5+log2x+33x+5(3x+5)=4⇒log3x+52x+3+log3x+53x+5+log2x+33x+5+log2x+33x+5=4    ∵logz(xy)=logzx+logzy⇒log3x+52x+3+1+log2x+33x+5+log2x+33x+5=4   ∵logaa=1⇒log3x+52x+3+log2x+33x+5+log2x+33x+5=3⇒log3x+52x+3+2log2x+33x+5=3⇒log3x+52x+3-1=2-2log2x+33x+5⇒log3x+52x+3-log3x+53x+5=21-log2x+33x+5    ∵logbb=1⇒log3x+52x+3-log3x+53x+5=2log2x+32x+3-log2x+33x+5⇒log3x+52x+33x+5=2log2x+32x+33x+5    ∵logbxy=logbx-logby⇒log2x+33x+5log3x+5=2log2x+33x+5log2x+3     ∵logb(a)=logalogb⇒log(2x+3)=2log3x+5⇒log2x+3=log3x+52⇒2x+3=3x+52⇒2x+3=9x2+25+30x⇒9x2+28x+22=0Discriminant,D=282-4922=784-792=-8<0So, No solution for x∈R
Regards