If `log{(x+y)/(3)}=((logx+logy))/(2)` , then `(x)/(y)+(y)/(x)` = …...

Answer

To solve the equation given in the problem, we start with the equation:

\[ \log\left(\frac{x+y}{3}\right) = \frac{\log x + \log y}{2} \]

### Step 1: Apply the properties of logarithms

Using the property of logarithms that states \(\log a + \log b = \log(ab)\), we can rewrite the right side:

\[ \frac{\log x + \log y}{2} = \log\left(\sqrt{xy}\right) \]

So, the equation becomes:

\[ \log\left(\frac{x+y}{3}\right) = \log\left(\sqrt{xy}\right) \]

### Step 2: Eliminate the logarithm

Since the logarithm function is one-to-one, we can equate the arguments:

\[ \frac{x+y}{3} = \sqrt{xy} \]

### Step 3: Cross-multiply to eliminate the fraction

Multiplying both sides by 3 gives:

\[ x + y = 3\sqrt{xy} \]

### Step 4: Square both sides

To eliminate the square root, we square both sides:

\[ (x + y)^2 = (3\sqrt{xy})^2 \]

This simplifies to:

\[ x^2 + 2xy + y^2 = 9xy \]

### Step 5: Rearrange the equation

Rearranging the equation gives:

\[ x^2 + y^2 + 2xy - 9xy = 0 \]

This simplifies to:

\[ x^2 + y^2 - 7xy = 0 \]

### Step 6: Factor the expression

We can express \(x^2 + y^2\) in terms of \(xy\):

\[ x^2 + y^2 = 7xy \]

### Step 7: Use the identity

We know that:

\[ \frac{x}{y} + \frac{y}{x} = \frac{x^2 + y^2}{xy} \]

Substituting the expression we found:

\[ \frac{x}{y} + \frac{y}{x} = \frac{7xy}{xy} = 7 \]

### Final Result

Thus, we conclude that:

\[ \frac{x}{y} + \frac{y}{x} = 7 \]