Answer:
3(x=y)
Step-by-step explanation:
Here, we have that
[tex]log2+\frac{1}{2}log (x)+ \frac{1}{2}log( y)=log(x+y)[/tex]
So, the goal here is to make the left-hand side look like the right-hand side.
On the right side, we have a single log with a coefficient of 1.
. So, we need to use some log properties to condense this log.
So, the first thing we will do is use this log properties fact.
[tex]rlog(x)=log(x^r)[/tex]
So that makes
[tex]\frac{1}{2} log(x)= log(x^\frac{1}{2} )\\Likewise,\\\frac{1}{2} log(y)= log(y^\frac{1}{2} )\\[/tex]
So we now have
[tex]log(2)+log(x^\frac{1}{2} )+ log(y^\frac{1}{2} )=log(x+y)[/tex]
Finally, using this property
[tex]log(x)+log(y)=log(xy)[/tex]
We have
[tex]log(2\sqrt{xy})=log(x+y)\\[/tex]
As you can see, there is no answer choice that matches this so let's continue to work.
Setting the coefficients equal to each other.
[tex]2\sqrt{xy} =(x+y)\\[/tex]
Squaring both sides gives us,
[tex]4xy=(x+y)^2\\4xy=x^2+2xy+y^2\\0=x^2-2xy+y^2\\0=(x-y)^2\\0=x-y\\x=y[/tex]
Let's verify this.
[tex]log2+0.5logx+0.5logx=log(2x)\\log2+0.5(logx^2)=log(2x)\\log(2)+log(x)=log(2x)\\log(2x)=log(2x)[/tex]
