If `5^(1+log_(4)x)+5^(-log_4x-1)=(26)/(5)`, then x=

Options

A. `4^(0)`
B. `4^(1)`
C. `4^(-1)`
D. `4^(-2)`

Answer

To solve the equation \( 5^{1 + \log_4 x} + 5^{-\log_4 x - 1} = \frac{26}{5} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 5^{1 + \log_4 x} + 5^{-\log_4 x - 1} = \frac{26}{5} \]
We can rewrite \( 5^{1 + \log_4 x} \) as \( 5 \cdot 5^{\log_4 x} \) and \( 5^{-\log_4 x - 1} \) as \( \frac{1}{5} \cdot 5^{-\log_4 x} \).

### Step 2: Substitute \( t = 5^{\log_4 x} \)
Let \( t = 5^{\log_4 x} \). Then, we can rewrite the equation as:
\[ 5t + \frac{1}{5t} = \frac{26}{5} \]

### Step 3: Multiply through by \( 5t \)
To eliminate the fraction, multiply through by \( 5t \):
\[ 25t^2 + 1 = 26t \]

### Step 4: Rearrange into standard quadratic form
Rearranging gives us:
\[ 25t^2 - 26t + 1 = 0 \]

### Step 5: Apply the quadratic formula
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 25 \), \( b = -26 \), and \( c = 1 \).
\[ t = \frac{26 \pm \sqrt{(-26)^2 - 4 \cdot 25 \cdot 1}}{2 \cdot 25} \]
Calculating the discriminant:
\[ t = \frac{26 \pm \sqrt{676 - 100}}{50} = \frac{26 \pm \sqrt{576}}{50} \]
\[ t = \frac{26 \pm 24}{50} \]

### Step 6: Find the values of \( t \)
Calculating the two possible values:
1. \( t = \frac{50}{50} = 1 \)
2. \( t = \frac{2}{50} = \frac{1}{25} \)

### Step 7: Relate back to \( x \)
Recall that \( t = 5^{\log_4 x} \):
1. For \( t = 1 \):
\[ 5^{\log_4 x} = 1 \implies \log_4 x = 0 \implies x = 4^0 = 1 \]
2. For \( t = \frac{1}{25} \):
\[ 5^{\log_4 x} = \frac{1}{25} \implies \log_4 x = -2 \implies x = 4^{-2} = \frac{1}{16} \]

### Final Answer
Thus, the values of \( x \) are:
\[ x = 1 \quad \text{and} \quad x = \frac{1}{16} \]