If `log_(5)((a+b)/(3))=(log_(5)a+log_(5)b)/(2),"then" (a^(4)+b^(4))/(a^(2)b^(2))=`

Answer

To solve the equation \( \log_5\left(\frac{a+b}{3}\right) = \frac{\log_5 a + \log_5 b}{2} \), we will follow these steps:

### Step 1: Rewrite the equation using properties of logarithms
We start with the given equation:
\[ \log_5\left(\frac{a+b}{3}\right) = \frac{\log_5 a + \log_5 b}{2} \]
Using the property of logarithms that states \( \log_b x + \log_b y = \log_b (xy) \), we can rewrite the right-hand side:
\[ \frac{\log_5 a + \log_5 b}{2} = \log_5\left(\sqrt{ab}\right) \]
Thus, we can rewrite the equation as:
\[ \log_5\left(\frac{a+b}{3}\right) = \log_5\left(\sqrt{ab}\right) \]

### Step 2: Eliminate the logarithm
Since the logarithms are equal, we can set the arguments equal to each other:
\[ \frac{a+b}{3} = \sqrt{ab} \]
Cross-multiplying gives:
\[ a + b = 3\sqrt{ab} \]

### Step 3: Square both sides
To eliminate the square root, we square both sides:
\[ (a + b)^2 = (3\sqrt{ab})^2 \]
This simplifies to:
\[ a^2 + 2ab + b^2 = 9ab \]

### Step 4: Rearranging the equation
Rearranging the equation gives:
\[ a^2 + b^2 + 2ab - 9ab = 0 \]
This simplifies to:
\[ a^2 + b^2 - 7ab = 0 \]

### Step 5: Express \(a^2 + b^2\) in terms of \(ab\)
From the equation above, we can express \(a^2 + b^2\) as:
\[ a^2 + b^2 = 7ab \]

### Step 6: Finding \(a^4 + b^4\)
To find \(a^4 + b^4\), we use the identity:
\[ a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2 \]
Substituting \(a^2 + b^2 = 7ab\):
\[ a^4 + b^4 = (7ab)^2 - 2a^2b^2 = 49a^2b^2 - 2a^2b^2 = 47a^2b^2 \]

### Step 7: Divide by \(a^2b^2\)
Now, we need to find:
\[ \frac{a^4 + b^4}{a^2b^2} = \frac{47a^2b^2}{a^2b^2} = 47 \]

Thus, the final answer is:
\[ \frac{a^4 + b^4}{a^2b^2} = 47 \]