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.