If `log_(5)x+log_(x)5=(5)/(2)`, then x= …....

Answer

To solve the equation \( \log_{5} x + \log_{x} 5 = \frac{5}{2} \), we can follow these steps:

### Step 1: Use the Change of Base Formula
We know that \( \log_{x} 5 = \frac{1}{\log_{5} x} \). Let's denote \( \log_{5} x = a \). Thus, we can rewrite the equation as:
\[ a + \frac{1}{a} = \frac{5}{2} \]

### Step 2: Multiply through by \( a \)
To eliminate the fraction, multiply both sides by \( a \):
\[ a^2 + 1 = \frac{5}{2} a \]

### Step 3: Rearrange the Equation
Rearranging gives us a standard quadratic equation:
\[ 2a^2 - 5a + 2 = 0 \]

### Step 4: Factor the Quadratic Equation
Now we will factor the quadratic equation:
\[ 2a^2 - 4a - a + 2 = 0 \]
Grouping the terms:
\[ 2a(a - 2) - 1(a - 2) = 0 \]
Factoring out \( (a - 2) \):
\[ (a - 2)(2a - 1) = 0 \]

### Step 5: Solve for \( a \)
Setting each factor to zero gives us:
1. \( a - 2 = 0 \) → \( a = 2 \)
2. \( 2a - 1 = 0 \) → \( a = \frac{1}{2} \)

### Step 6: Substitute Back to Find \( x \)
Recall that \( a = \log_{5} x \). We can substitute back to find \( x \):
1. For \( a = 2 \):
\[ \log_{5} x = 2 \implies x = 5^2 = 25 \]
2. For \( a = \frac{1}{2} \):
\[ \log_{5} x = \frac{1}{2} \implies x = 5^{\frac{1}{2}} = \sqrt{5} \]

### Final Answer
Thus, the values of \( x \) are:
\[ x = 25 \quad \text{and} \quad x = \sqrt{5} \]

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