The number of solutions of the equation `5^x+5^-x=log_(10) 25, x in R` is

Options

A. 0
B. 1
C. 2
D. infinitely many

Explanation

To solve the equation \( 5^x + 5^{-x} = \log_{10} 25 \), we will analyze both sides of the equation step by step.

### Step 1: Rewrite the equation
We start with the equation:
\[ 5^x + 5^{-x} = \log_{10} 25 \]

### Step 2: Simplify the left-hand side
Let \( t = 5^x \). Then, \( 5^{-x} = \frac{1}{t} \). Substituting these into the equation gives:
\[ t + \frac{1}{t} = \log_{10} 25 \]

### Step 3: Analyze the left-hand side
The expression \( t + \frac{1}{t} \) can be analyzed using the AM-GM inequality:
\[ t + \frac{1}{t} \geq 2 \]
This means that the minimum value of \( t + \frac{1}{t} \) is 2, which occurs when \( t = 1 \) (or \( x = 0 \)).

### Step 4: Simplify the right-hand side
Now, we simplify the right-hand side:
\[ \log_{10} 25 = \log_{10} (5^2) = 2 \log_{10} 5 \]
Since \( \log_{10} 5 \) is less than 1 (because 5 is less than 10), we have:
\[ 2 \log_{10} 5 < 2 \]

### Step 5: Compare both sides
From our analysis:
- The left-hand side \( t + \frac{1}{t} \) is always greater than or equal to 2.
- The right-hand side \( \log_{10} 25 \) is less than 2.

### Step 6: Conclusion
Since the left-hand side can never be equal to the right-hand side (the left-hand side is always at least 2, while the right-hand side is less than 2), we conclude that there are no solutions to the equation.

Thus, the number of solutions of the equation \( 5^x + 5^{-x} = \log_{10} 25 \) is:
\[ \boxed{0} \]