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

1 Answer
Aug 15, 2017

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

Explanation:

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

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

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

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

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

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

or y^2-6y+2=0

and using quadratic formula

y=(6+-sqrt(36-8))/2=3+-sqrt7

If y=3+sqrt7, x=-2-sqrt7, but we cannot have x<0

and if y=3-sqrt7, x=-2+sqrt7

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