Solve the simultaneous equations log_a(x+y) = 0loga(x+y)=0 and 2log_ax = log(4y +1)2logax=log(4y+1) for xx and yy ?

1 Answer
Aug 15, 2017

Solution is (-2+sqrt7,3-sqrt7)(2+7,37)

Explanation:

As log_a(x+y)=0loga(x+y)=0, x+y=1x+y=1 as log of 11 is always zero. (A)

Further 2log_ax=log_a(4y+1)=>log_ax^2=log_a(4y+1)2logax=loga(4y+1)logax2=loga(4y+1) (B)

i.e. x^2=4y+1x2=4y+1

as from A, x=1-yx=1y, putting this in B, we get

(1-y)^2=4y+1(1y)2=4y+1

or 1-2y+y^2=4y+112y+y2=4y+1

or y^2-6y+2=0y26y+2=0

and using quadratic formula

y=(6+-sqrt(36-8))/2=3+-sqrt7y=6±3682=3±7

If y=3+sqrt7y=3+7, x=-2-sqrt7x=27, but we cannot have x<0x<0

and if y=3-sqrt7y=37, x=-2+sqrt7x=2+7

Hence solution is (-2+sqrt7,3-sqrt7)(2+7,37)