Let a and b two real numbers positives and different than 1. What is the relation between a and b so that the equation x^2-x(log(b)a)+2log(a)b=0 will have two equal roots

The answer is a^2=b

To solve it i used the delta expression of the baskhara formula: b^2-4*a*c since it needs to have the same the same two roots it means delta=0

So I applied the formula and got to log^2(a)b-8log(b)a=0 ==> [log(a)b]^2=8log(b)a

- taking the SQUARE root of both sides i got log(a)b=log(b)a

- I can apply change of base to log(b)a and the end result will be: log(a)b=1/log(a)b

- multiplying both sides by log(a)b I got [log(a)b]^2=1

-I took the square root of both since sqrt(1) =1 and got : log (a)b=1

- applying a consequence of the definition of logarithm(log(a)a=1) I got: log(a)b=log(a)a so b=a?

Can someone kindly point me in the right direction I must have make a mistake somewhere.

I am new in the topic of logarithms.

Your help is deeply appreciated.

thank You.