MATH SOLVE

4 months ago

Q:
# ln ( (x+y)/3 ) =( ln(x)+ln(y))/2
Calculate x/y +y/x

Accepted Solution

A:

ln((x + y)/3) = (ln(x) + ln(y))/2
First, we'll get rid of the natural logarithm by exponentiating both sides of the equation:
e^(ln((x + y)/3)) = e^((ln(x) + ln(y))/2)
The left side simplifies to:
(x + y)/3 = e^((ln(x) + ln(y))/2)
Now, we can eliminate the exponent on the right side by squaring both sides of the equation:
((x + y)/3)^2 = e^(ln(x) + ln(y))
Now, let's use the properties of exponents and the fact that e^(ln(x)) = x:
((x + y)/3)^2 = x * y
Next, we can expand the left side:
(x^2 + 2xy + y^2)/9 = x * y
Now, let's multiply both sides of the equation by 9 to get rid of the denominator:
x^2 + 2xy + y^2 = 9xy
Now, move all the terms to one side of the equation:
x^2 - 7xy + y^2 = 0
We have a quadratic equation in terms of x and y. To simplify it further, we can factor it:
(x - y)(x - 6y) = 0
Now, we have two possible solutions:
x - y = 0
x - 6y = 0
Let's solve each equation separately:
x - y = 0
x = y
x - 6y = 0
x = 6y
Now, we can calculate x/y + y/x for each case:
x/y + y/x when x = y:
(x/y + y/x) = (y/y + y/y) = (1 + 1) = 2
x/y + y/x when x = 6y:
(x/y + y/x) = (6y/y + y/6y) = (6 + 1/6) = 37/6
So, there are two possible values for x/y + y/x: 2 and 37/6, depending on the values of x and y.